
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 regular 100-sided polygon is placed on a table, with the numbers $1, 2, \ldots, 100$ written at its vertices. These numbers are then rewritten in order of their distance from the front edge of the table. If two vertices are at an equal distance from the edge, the left number is listed first, followed by the right number. Form all possible sets of numbers corresponding to different positions of the 100-sided polygon. Calculate the sum of the numbers that occupy the 13th position from the left in these sets.

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