
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
We arrange several "0"s and "1"s in a row, defining the capacity of this arrangement as the number of times "10" appears in the arrangement. For example, the capacity of "1010" is 2, and the capacity of "0101" is 1. If we arrange 50 "0"s and 50 "1"s in a row, we will get $\binom{100}{50}$ different arrangements. Find the sum of the capacities of these arrangements.

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