Question 117
Let $a_0 = 2^{2025}$. Two players take turns extending a sequence $a_0, a_1, a_2, \dots$ by the following rules:
On the $n$-th move, the player chooses $a_n = a_{n-1} + 1$ or $a_n = S(a_{n-1})$, where $S(x)$ denotes the sum of the digits of $x$.
The game ends when either three identical numbers appear in the sequence, or four consecutive terms form an arithmetic progression. The player who makes the final move (i.e., causes the game to end) wins.
Assuming both players play optimally, determine which player has a winning strategy.
(Batbayasgalan Balkhuu)
On the $n$-th move, the player chooses $a_n = a_{n-1} + 1$ or $a_n = S(a_{n-1})$, where $S(x)$ denotes the sum of the digits of $x$.
The game ends when either three identical numbers appear in the sequence, or four consecutive terms form an arithmetic progression. The player who makes the final move (i.e., causes the game to end) wins.
Assuming both players play optimally, determine which player has a winning strategy.
(Batbayasgalan Balkhuu)