
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 an odd prime number. For positive integer  $k$  satisfying  $1\le k\le p-1$ , the number of divisors of  $kp+1$  between  $k$  and  $p$  exclusive is  $a_k$ . Find the value of  $a_1+a_2+\ldots + a_{p-1}$ .

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