
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
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Square $PQRS$ has center $O$, and side length $PS=800$. Points $G$ and $H$ are on side $PS$, with $PG < QH$ and $G$ between $P$ and $H$. Angle $\angle GOH = 60^\circ$ and $GH = 350$. It is given that $QH = a + b\sqrt{c}$, where $a, b$, and $c$ are positive integers, and $c$ is not divisible by the square of any prime, determine $a+b+c$.

**

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