
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
Given that in $\triangle DEF$, $DE = 45$ and $DF = 75$, and a circle with center $D$ and radius $DE$ intersects $\overline{EF}$ at points $E$ and $Y$, with $\overline{EY}$ and $\overline{FY}$ having integer lengths, calculate the length of $EF$.

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