
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
If  $p=2^{16}+1$  is a prime, find the maximum possible number of elements in a set  $S$  of positive integers less than  $p$  so no two distinct  $a,b$  in  $S$  satisfy   $$ a^2\equiv b\pmod{p}. $$  

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