
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  $x_1, x_2, x_3, . . .$  is defined by  $x_1 = 2022$  and  $x_{n+1}= 7x_n + 5$  for all positive integers  $n$ . Determine the maximum positive integer  $m$  such that   $$ \frac{x_n(x_n - 1)(x_n - 2) . . . (x_n - m + 1)}{m!} $$   is never a multiple of  $7$  for any positive integer  $n$ .

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