
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
Let  $ABCDEFGHIJ$  be a regular  $10$ -sided polygon that has all its vertices in one circle with center  $O$  and radius  $5$ . The diagonals  $AD$  and  $BE$  intersect at  $P$  and the diagonals  $AH$  and  $BI$  intersect at  $Q$ . Calculate the measure of the  segment  $PQ$ .

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