
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
Given a regular hexagon $ABCDEF$ with center $O$. Let $P$, $Q$, and $R$ be the perpendiculars dropped from vertex $A$ onto sides $CD$, $DE$, and $EF$, respectively. Assume that $OP=2$. Find the sum $AP + AQ + AR$.

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