
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
In a square $PQRS$ with center $M$ and side length $800$, points $J$ and $K$ are on side $PQ$ such that $J$ is nearer to $P$ than $K$, and $PJ < QK$. If $m\angle JMK = 30^\circ$ and $JK = 350$, determine $QK$ expressed in the form $v + w\sqrt{u}$ where $v$, $w$, $u$ are positive integers and $u$ is not divisible by the square of any prime. Find $v + w + u$.

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