
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
A group of 101 Dalmatians participate in an election, where they each vote independently for either candidate \( A \) or \( B \) with equal probability. If \( X \) Dalmatians voted for the winning candidate, the expected value of \( X^{2} \) can be expressed as \( \frac{a}{b} \) for positive integers \( a, b \) with \( \operatorname{gcd}(a, b)=1 \). Find the unique positive integer \( k \leq 103 \) such that \( 103 \mid a - b k \).

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