
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 square $PQRS$ with center $T$, side $PQ = 800$, points $M$ and $N$ are on $PQ$ such that $PM < NQ$ and $M$ is between $P$ and $N$. If $m\angle MTN = 45^\circ$ and $MN = 300$, and if $NQ = a + b\sqrt{c}$, where $a, b, c$ are positive integers with $c$ not divisible by the square of any prime, find $a+b+c$.

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