
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 $EFGH$ be an isosceles trapezoid with bases $EF=100$ and $GH=25$. Suppose $EG=FH=y$ and a circle with center on $\overline{EF}$ is tangent to segments $\overline{EG}$ and $\overline{FH}$. If $n$ is the smallest possible value of $y$, then calculate $n^2$.

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