
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 \( \mathrm{p} \) be an odd prime number. Consider the set \(\{ 1, 2, \ldots, 2\mathrm{p} \}\) and find the number of subsets \( A \) that satisfy the following conditions: 
1. \( A \) contains exactly \( \mathrm{p} \) elements.
2. The sum of all elements in \( A \) is divisible by \( \mathrm{p} \).

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