
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 $O$, and side length $AB=1200$. Points $G$ and $H$ are on side $AB$, with $AG < BH$ and $G$ between $A$ and $H$. If $\angle GOH = 60^\circ$ and $GH = 600$, find the length of $BH$ expressed in the form $p + q\sqrt{r}$, where $p, q,$ and $r$ are integers, and $r$ is not divisible by the square of any prime. Calculate $p+q+r$.

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