
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
A trapezoid has one base that is 150 units longer than its other base. The segment that joins the midpoints of the legs divides the trapezoid into two regions whose areas are in the ratio $3: 4$. Let $x$ be the length of the segment of the trapezoid that is parallel to the bases and that divides the trapezoid into two equal-area regions. Find the greatest integer that does not exceed $x^3/1000$.

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