
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. Consider the set \(\{1,2, \cdots, 2p\}\) and its subset \(A\) satisfying the following two conditions:
1. \(|A|=p\);
2. The sum of all elements in \(A\) is divisible by \(p\).

Find the number of such subsets \(A\).

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