
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 a given odd prime. Find the largest integer  $k'$  such that it is possible to partition  $\{1,2,\cdots,p-1\}$  into two sets  $X,Y$  such that for any  $k$  with  $0 \le k \le k'$ ,  $$ \sum_{a \in X}a^k \equiv \sum_{b \in Y}b^k \pmod p $$  *houkai*

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