
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
At a recent math contest, Evan was asked to find  $2^{2016} \pmod{p}$  for a given prime number  $p$  with  $100 < p < 500$ . Evan has forgotten what the prime  $p$  was, but still remembers how he solved it:


- Evan first tried taking  $2016$  modulo  $p - 1$ , but got a value  $e$  larger than  $100$ .
- However, Evan noted that  $e - \frac{1}{2}(p - 1) = 21$ , and then realized the answer was  $-2^{21} \pmod{p}$ .

What was the prime  $p$ ?

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