
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
Square $ABCD$ has center $O$, with side length $AB = 1200$. Points $P$ and $Q$ lie on side $AB$, with $AP < BQ$ and $P$ between $A$ and $Q$, $m\angle POQ = 30^\circ$, and $PQ = 500$. Given that $BQ = p + q\sqrt{r}$, where $p, q$, and $r$ are integers and $r$ is not divisible by the square of any prime, find $p+q+r$.

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