
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]


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From an arbitrary internal point \( O \) of a convex \( n \)-gon, perpendiculars are dropped to its sides (or their extensions). On each perpendicular, a vector is constructed from point \( O \) towards the side, the length of which is equal to half the length of the side on which the perpendicular is dropped. Determine the sum of the constructed vectors.

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