
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  $a>1$  be a positive integer.  The sequence of natural numbers  $\{a_n\}_{n\geq 1}$  is defined such that  $a_1 = a$  and for all  $n\geq 1$ ,  $a_{n+1}$  is the largest prime factor of  $a_n^2 - 1$ .  Determine the smallest possible value of  $a$  such that the numbers  $a_1$ ,  $a_2$ , $\ldots$ ,  $a_7$  are all distinct.

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