
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  $p = 101$  and let  $S$  be the set of  $p$ -tuples  $(a_1, a_2, \dots, a_p) \in \mathbb{Z}^p$  of integers. Let  $N$  denote the number of functions  $f: S \to \{0, 1, \dots, p-1\}$  such that

	
- $f(a + b) + f(a - b) \equiv 2\big(f(a) + f(b)\big) \pmod{p}$  for all  $a, b \in S$ , and
- $f(a) = f(b)$  whenever all components of  $a-b$  are divisible by  $p$ .

Compute the number of positive integer divisors of  $N$ . (Here addition and subtraction in  $\mathbb{Z}^p$  are done component-wise.)

*Proposed by Ankan Bhattacharya*

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