
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Let $p$ be a prime number. Tristan and Abigaëlle play the following game. Tristan writes an integer $X \geq 1$ on the board and gives Abigaëlle a sequence $\left(a_{n}\right)_{n \in \mathbb{N}}$ of strictly positive integers. Abigaëlle then plays an infinite number of turns. On the $n^{\text{th}}$ turn of the game,

Abigaëlle replaces, at her discretion, the integer $Y$ written on the board with either $Y + a_{n}$ or $Y \cdot a_{n}$.

Abigaëlle wins if, after a finite number of turns, she manages to write a multiple of $p$ on the board. Determine if she can always win regardless of Tristan's initial choices, in each of the following cases:
a) $p = 10^{9} + 7$;
b) $p = 10^{9} + 9$.

Remark: It is assumed that $10^{9} + 7$ and $10^{9} + 9$ are prime.

Present the answer in LaTex format: \boxed{Your answer}
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