Number Sequence Game (noch nicht übersetzt)

The number sequence game starts with a sequence S of N numbers written on a line.

Two players alternate turns. The players on their respective turns must select and remove either the first or the last number remaining in the sequence.

A player's own score is (determined by) the sum of all the numbers that player has taken. Each player attempts to maximize their own sum.

If N = 4 and S = {1, 2, 10, 3}, then each player maximizes their own score as follows:

• Player 1: removes the first number (1)
• Player 2: removes the last number from the remaining sequence (3)
• Player 1: removes the last number from the remaining sequence (10)
• Player 2: removes the remaining number (2)

Player 1 score is 1 + 10 = 11.

Let F(N) be the score of player 1 if both players follow the optimal strategy for the sequence S = {s₁, s₂, ..., s_N} defined as:

• s₁ = 0
• s_i+1 = (s_i² + 45) modulo 1 000 000 007

The sequence begins with S = {0, 45, 2070, 4284945, 753524550, 478107844, 894218625, ...}.

You are given F(2) = 45, F(4) = 4284990, F(100) = 26365463243, F(10⁴) = 2495838522951.

Find F(10⁸).