
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
Triangle $XYZ$ is a right triangle with $XZ = 10,$ $YZ = 26,$ and right angle at $Z.$ Point $N$ is the midpoint of $XY,$ and $E$ is on the same side of line $XY$ as $Z$ such that $XE = YE = 20.$ Calculate the area of triangle $ZEN$ and express it as $\frac {u\sqrt {v}}{w},$ where $u,$ $v,$ and $w$ are positive integers with $u$ and $w$ relatively prime, and $v$ is not divisible by the square of any prime. Find $u + v + w.$

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