
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 the xy-plane, consider a right triangle $ABC$ with the right angle at $C$. The hypotenuse $AB$ is of length $50$. The medians through vertices $A$ and $B$ are described by the lines $y = x + 5$ and $y = 2x + 2$, respectively. Determine the area of triangle $ABC$.

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