
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
Roger initially has 20 socks in a drawer, each of which is either white or black. He chooses a sock uniformly at random from the drawer and throws it away. He repeats this action until there are equal numbers of white and black socks remaining. Suppose that the probability he stops before all socks are gone is \( p \). If the sum of all distinct possible values of \( p \) over all initial combinations of socks is \(\frac{a}{b}\) for relatively prime positive integers \( a \) and \( b \), compute \( 100a + b \).

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