
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
Two bases of a trapezoid have a length difference of $50$ units. The segment that joins the midpoints of the legs divides the trapezoid into two small trapezoids whose area ratio is $3 : 5$. If $x$ is the segment length parallel to the bases and divides the trapezoid into two regions of equal area, compute the largest integer that is less than $x^2/50$.

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