
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 $ABC$ be an isosceles triangle in the xy-plane with its vertex $C$ at the origin. The hypotenuse $AB$ has a length of $50$. The median through vertex $A$ lies along the line $y=x+3$, and the median through vertex $B$ lies along $y=-x+3$. Determine the area of triangle $ABC$.

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