
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
The sequence \(x_{1}, x_{2}, x_{3}, \ldots\) is defined by \(x_{1}=2022\) and \(x_{n+1}=7 x_{n}+5\) for all positive integers \(n\). Determine the maximum positive integer \(m\) such that
\[
\frac{x_{n}(x_{n}-1)(x_{n}-2)\ldots(x_{n}-m+1)}{m!}
\]
is never a multiple of 7 for any positive integer \(n\).

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