
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 fixed prime number. Consider four integers $a, b, c, d \in \mathbb{Z}$ and a sequence $\left(x_{n}\right)$ such that:

$$
\begin{cases}
x_{0}=c \\
x_{1}=d \\
x_{n+2}=a x_{n+1}+b x_{n} \quad \text{for} \, n \geq 0
\end{cases}
$$

Let $t_{p}$ be the smallest integer such that $x_{n+t_{p}} \equiv x_{n} \pmod{p}$ for all $n \geq p^{2}$. (We set $t_{p}=+\infty$ if no such integer exists). What is the largest possible value of $t_{p}$?

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