
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
Find all integers  $a$  such that there is a prime number of  $p\ge  5$  that divides
  ${p-1 \choose 2}$   $+ {p-1 \choose 3} a$   $+{p-1 \choose 4} a^2$ + ...+ $ {p-1 \choose p-3} a^{p-5} .$ 

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