
"""simple docstring"""
import math
import warnings
from dataclasses import dataclass
from typing import List, Optional, Tuple, Union

import numpy as np
import torch
from scipy import integrate

from ..configuration_utils import ConfigMixin, register_to_config
from ..utils import BaseOutput
from .scheduling_utils import KarrasDiffusionSchedulers, SchedulerMixin
@dataclass
# Copied from diffusers.schedulers.scheduling_ddpm.DDPMSchedulerOutput with DDPM->LMSDiscrete
class      lowerCAmelCase_						(							A__				):



							'''simple docstring'''





							_snake_case		     =						42
							_snake_case		     =						None

def 				lowercase      (_lowerCAmelCase				,  _lowerCAmelCase=0.999				,  _lowerCAmelCase="cosine"				,  ):
						if alpha_transform_type == "cosine":

												def alpha_bar_fn(_lowerCAmelCase		):
																		return math.cos((t + 0.008) / 1.008 * math.pi / 2		) ** 2

						elif alpha_transform_type == "exp":

												def alpha_bar_fn(_lowerCAmelCase		):
																		return math.exp(t * -12.0		)

						else:
												raise ValueError(f"""Unsupported alpha_tranform_type: {alpha_transform_type}"""		)

						__lowerCAmelCase        =       []
						for i in range(_lowerCAmelCase		):
												__lowerCAmelCase        =       i / num_diffusion_timesteps
												__lowerCAmelCase        =       (i + 1) / num_diffusion_timesteps
												betas.append(min(1 - alpha_bar_fn(_lowerCAmelCase		) / alpha_bar_fn(_lowerCAmelCase		)				,  _lowerCAmelCase		)		)
						return torch.tensor(_lowerCAmelCase				,  dtype=torch.floataa		)



class      lowerCAmelCase_						(							A__							,							A__				):



							'''simple docstring'''





							_snake_case		     =						[e.name for e in KarrasDiffusionSchedulers]
							_snake_case		     =						1




							@register_to_config
							def __init__(				self  ,							snake_case_ = 1_000  ,							snake_case_ = 0.0_001  ,							snake_case_ = 0.02  ,							snake_case_ = "linear"  ,							snake_case_ = None  ,							snake_case_ = False  ,							snake_case_ = "epsilon"  ,							snake_case_ = "linspace"  ,							snake_case_ = 0  ,							)   ->       Optional[Any]:
													if trained_betas is not None:
																			__lowerCAmelCase        =       torch.tensor(snake_case_  ,							dtype=torch.floataa							)
													elif beta_schedule == "linear":
																			__lowerCAmelCase        =       torch.linspace(snake_case_  ,							snake_case_  ,							snake_case_  ,							dtype=torch.floataa							)
													elif beta_schedule == "scaled_linear":
																			# this schedule is very specific to the latent diffusion model.
																			__lowerCAmelCase        =       (
																			    torch.linspace(beta_start**0.5  ,							beta_end**0.5  ,							snake_case_  ,							dtype=torch.floataa							) ** 2
																			)
													elif beta_schedule == "squaredcos_cap_v2":
																			# Glide cosine schedule
																			__lowerCAmelCase        =       betas_for_alpha_bar(snake_case_							)
													else:
																			raise NotImplementedError(f"""{beta_schedule} does is not implemented for {self.__class__}"""							)

													__lowerCAmelCase        =       1.0 - self.betas
													__lowerCAmelCase        =       torch.cumprod(self.alphas  ,							dim=0							)

													__lowerCAmelCase        =       np.array(((1 - self.alphas_cumprod) / self.alphas_cumprod) ** 0.5							)
													__lowerCAmelCase        =       np.concatenate([sigmas[::-1], [0.0]]							).astype(np.floataa							)
													__lowerCAmelCase        =       torch.from_numpy(snake_case_							)

													# setable values
													__lowerCAmelCase        =       None
													__lowerCAmelCase        =       use_karras_sigmas
													self.set_timesteps(snake_case_  ,							snake_case_							)
													__lowerCAmelCase        =       []
													__lowerCAmelCase        =       False




							@property
							def  A__		(				self							)   ->       Any:
													# standard deviation of the initial noise distribution
													if self.config.timestep_spacing in ["linspace", "trailing"]:
																			return self.sigmas.max()

													return (self.sigmas.max() ** 2 + 1) ** 0.5




							def  A__		(				self  ,							snake_case_  ,							snake_case_							)   ->       torch.FloatTensor:
													if isinstance(snake_case_  ,							torch.Tensor							):
																			__lowerCAmelCase        =       timestep.to(self.timesteps.device							)
													__lowerCAmelCase        =       (self.timesteps == timestep).nonzero().item()
													__lowerCAmelCase        =       self.sigmas[step_index]
													__lowerCAmelCase        =       sample / ((sigma**2 + 1) ** 0.5)
													__lowerCAmelCase        =       True
													return sample




							def  A__		(				self  ,							snake_case_  ,							snake_case_  ,							snake_case_							)   ->       Optional[int]:

													def lms_derivative(snake_case_							):
																			__lowerCAmelCase        =       1.0
																			for k in range(snake_case_							):
																									if current_order == k:
																															continue
																									prod *= (tau - self.sigmas[t - k]) / (self.sigmas[t - current_order] - self.sigmas[t - k])
																			return prod

													__lowerCAmelCase        =       integrate.quad(snake_case_  ,							self.sigmas[t]  ,							self.sigmas[t + 1]  ,							epsrel=1e-4							)[0]

													return integrated_coeff




							def  A__		(				self  ,							snake_case_  ,							snake_case_ = None							)   ->       List[str]:
													__lowerCAmelCase        =       num_inference_steps

													# "linspace", "leading", "trailing" corresponds to annotation of Table 2. of https://arxiv.org/abs/2305.08891
													if self.config.timestep_spacing == "linspace":
																			__lowerCAmelCase        =       np.linspace(0  ,							self.config.num_train_timesteps - 1  ,							snake_case_  ,							dtype=snake_case_							)[
																			    ::-1
																			].copy()
													elif self.config.timestep_spacing == "leading":
																			__lowerCAmelCase        =       self.config.num_train_timesteps // self.num_inference_steps
																			# creates integer timesteps by multiplying by ratio
																			# casting to int to avoid issues when num_inference_step is power of 3
																			__lowerCAmelCase        =       (np.arange(0  ,							snake_case_							) * step_ratio).round()[::-1].copy().astype(snake_case_							)
																			timesteps += self.config.steps_offset
													elif self.config.timestep_spacing == "trailing":
																			__lowerCAmelCase        =       self.config.num_train_timesteps / self.num_inference_steps
																			# creates integer timesteps by multiplying by ratio
																			# casting to int to avoid issues when num_inference_step is power of 3
																			__lowerCAmelCase        =       (np.arange(self.config.num_train_timesteps  ,							0  ,							-step_ratio							)).round().copy().astype(snake_case_							)
																			timesteps -= 1
													else:
																			raise ValueError(
																			    f"""{self.config.timestep_spacing} is not supported. Please make sure to choose one of 'linspace', 'leading' or 'trailing'."""							)

													__lowerCAmelCase        =       np.array(((1 - self.alphas_cumprod) / self.alphas_cumprod) ** 0.5							)
													__lowerCAmelCase        =       np.log(snake_case_							)
													__lowerCAmelCase        =       np.interp(snake_case_  ,							np.arange(0  ,							len(snake_case_							)							)  ,							snake_case_							)

													if self.use_karras_sigmas:
																			__lowerCAmelCase        =       self._convert_to_karras(in_sigmas=snake_case_							)
																			__lowerCAmelCase        =       np.array([self._sigma_to_t(snake_case_  ,							snake_case_							) for sigma in sigmas]							)

													__lowerCAmelCase        =       np.concatenate([sigmas, [0.0]]							).astype(np.floataa							)

													__lowerCAmelCase        =       torch.from_numpy(snake_case_							).to(device=snake_case_							)
													if str(snake_case_							).startswith("""mps"""							):
																			# mps does not support float64
																			__lowerCAmelCase        =       torch.from_numpy(snake_case_							).to(snake_case_  ,							dtype=torch.floataa							)
													else:
																			__lowerCAmelCase        =       torch.from_numpy(snake_case_							).to(device=snake_case_							)

													__lowerCAmelCase        =       []




							def  A__		(				self  ,							snake_case_  ,							snake_case_							)   ->       List[str]:
													# get log sigma
													__lowerCAmelCase        =       np.log(snake_case_							)

													# get distribution
													__lowerCAmelCase        =       log_sigma - log_sigmas[:, np.newaxis]

													# get sigmas range
													__lowerCAmelCase        =       np.cumsum((dists >= 0)  ,							axis=0							).argmax(axis=0							).clip(max=log_sigmas.shape[0] - 2							)
													__lowerCAmelCase        =       low_idx + 1

													__lowerCAmelCase        =       log_sigmas[low_idx]
													__lowerCAmelCase        =       log_sigmas[high_idx]

													# interpolate sigmas
													__lowerCAmelCase        =       (low - log_sigma) / (low - high)
													__lowerCAmelCase        =       np.clip(snake_case_  ,							0  ,							1							)

													# transform interpolation to time range
													__lowerCAmelCase        =       (1 - w) * low_idx + w * high_idx
													__lowerCAmelCase        =       t.reshape(sigma.shape							)
													return t




							def  A__		(				self  ,							snake_case_							)   ->       torch.FloatTensor:
													__lowerCAmelCase        =       in_sigmas[-1].item()
													__lowerCAmelCase        =       in_sigmas[0].item()

													__lowerCAmelCase        =       7.0  # 7.0 is the value used in the paper
													__lowerCAmelCase        =       np.linspace(0  ,							1  ,							self.num_inference_steps							)
													__lowerCAmelCase        =       sigma_min ** (1 / rho)
													__lowerCAmelCase        =       sigma_max ** (1 / rho)
													__lowerCAmelCase        =       (max_inv_rho + ramp * (min_inv_rho - max_inv_rho)) ** rho
													return sigmas




							def  A__		(				self  ,							snake_case_  ,							snake_case_  ,							snake_case_  ,							snake_case_ = 4  ,							snake_case_ = True  ,							)   ->       Union[LMSDiscreteSchedulerOutput, Tuple]:
													if not self.is_scale_input_called:
																			warnings.warn(
																			    """The `scale_model_input` function should be called before `step` to ensure correct denoising. """
																			    """See `StableDiffusionPipeline` for a usage example."""							)

													if isinstance(snake_case_  ,							torch.Tensor							):
																			__lowerCAmelCase        =       timestep.to(self.timesteps.device							)
													__lowerCAmelCase        =       (self.timesteps == timestep).nonzero().item()
													__lowerCAmelCase        =       self.sigmas[step_index]

													# 1. compute predicted original sample (x_0) from sigma-scaled predicted noise
													if self.config.prediction_type == "epsilon":
																			__lowerCAmelCase        =       sample - sigma * model_output
													elif self.config.prediction_type == "v_prediction":
																			# * c_out + input * c_skip
																			__lowerCAmelCase        =       model_output * (-sigma / (sigma**2 + 1) ** 0.5) + (sample / (sigma**2 + 1))
													elif self.config.prediction_type == "sample":
																			__lowerCAmelCase        =       model_output
													else:
																			raise ValueError(
																			    f"""prediction_type given as {self.config.prediction_type} must be one of `epsilon`, or `v_prediction`"""							)

													# 2. Convert to an ODE derivative
													__lowerCAmelCase        =       (sample - pred_original_sample) / sigma
													self.derivatives.append(snake_case_							)
													if len(self.derivatives							) > order:
																			self.derivatives.pop(0							)

													# 3. Compute linear multistep coefficients
													__lowerCAmelCase        =       min(step_index + 1  ,							snake_case_							)
													__lowerCAmelCase        =       [self.get_lms_coefficient(snake_case_  ,							snake_case_  ,							snake_case_							) for curr_order in range(snake_case_							)]

													# 4. Compute previous sample based on the derivatives path
													__lowerCAmelCase        =       sample + sum(
													    coeff * derivative for coeff, derivative in zip(snake_case_  ,							reversed(self.derivatives							)							)							)

													if not return_dict:
																			return (prev_sample,)

													return LMSDiscreteSchedulerOutput(prev_sample=snake_case_  ,							pred_original_sample=snake_case_							)




							def  A__		(				self  ,							snake_case_  ,							snake_case_  ,							snake_case_  ,							)   ->       torch.FloatTensor:
													# Make sure sigmas and timesteps have the same device and dtype as original_samples
													__lowerCAmelCase        =       self.sigmas.to(device=original_samples.device  ,							dtype=original_samples.dtype							)
													if original_samples.device.type == "mps" and torch.is_floating_point(snake_case_							):
																			# mps does not support float64
																			__lowerCAmelCase        =       self.timesteps.to(original_samples.device  ,							dtype=torch.floataa							)
																			__lowerCAmelCase        =       timesteps.to(original_samples.device  ,							dtype=torch.floataa							)
													else:
																			__lowerCAmelCase        =       self.timesteps.to(original_samples.device							)
																			__lowerCAmelCase        =       timesteps.to(original_samples.device							)

													__lowerCAmelCase        =       [(schedule_timesteps == t).nonzero().item() for t in timesteps]

													__lowerCAmelCase        =       sigmas[step_indices].flatten()
													while len(sigma.shape							) < len(original_samples.shape							):
																			__lowerCAmelCase        =       sigma.unsqueeze(-1							)

													__lowerCAmelCase        =       original_samples + noise * sigma
													return noisy_samples




							def __len__(				self							)   ->       Tuple:
													return self.config.num_train_timesteps


"""simple docstring"""
from __future__ import annotations
class      lowerCAmelCase_						:



							'''simple docstring'''





							def __init__(				self  ,							snake_case_							)   ->       None:
													__lowerCAmelCase        =       data
													__lowerCAmelCase        =       None
class      lowerCAmelCase_						:



							'''simple docstring'''





							def __init__(				self							)   ->       List[str]:
													__lowerCAmelCase        =       None




							def  A__		(				self  ,							snake_case_							)   ->       int:
													__lowerCAmelCase        =       Node(snake_case_							)
													__lowerCAmelCase        =       self.head
													__lowerCAmelCase        =       new_node
													return self.head.data




							def  A__		(				self							)   ->       int | None:
													__lowerCAmelCase        =       self.head
													__lowerCAmelCase        =       self.head
													if self.head:
																			while fast_pointer and fast_pointer.next:
																									__lowerCAmelCase        =       fast_pointer.next.next
																									__lowerCAmelCase        =       slow_pointer.next
																			return slow_pointer.data
													else:
																			print("""No element found."""							)
																			return None


if __name__ == "__main__":
			SCREAMING_SNAKE_CASE_			      =					LinkedList()
			for _ in range(int(input().strip())):
						SCREAMING_SNAKE_CASE_			      =					int(input().strip())
						link.push(data)
			print(link.middle_element())
