
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
Let triangle $XYZ$ be a right triangle in the xy-plane with a right angle at $Z$. Given that the length of the hypotenuse $XY$ is $50$, and that the medians through $X$ and $Y$ lie along the lines $y=x+1$ and $y=2x+2$ respectively, find the area of triangle $XYZ$.

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