
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 number. Consider the set \(\{1,2, \cdots, 2p\}\) and a subset \( A \) that satisfies the following two conditions:
1. \( A \) contains exactly \( p \) elements;
2. The sum of all elements in \( A \) is divisible by \( p \).

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

(36th International Mathematical Olympiad, 1995)

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