
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
In right $\triangle DEF$ with hypotenuse $\overline{DE}$, $DF = 9$, $EF = 40$, and $\overline{FG}$ is the altitude to $\overline{DE}$. Let $\omega'$ be the circle having $\overline{FG}$ as a diameter. Let $J$ be a point outside $\triangle DEF$ such that $\overline{DJ}$ and $\overline{EJ}$ are both tangent to circle $\omega'$. Find the ratio of the perimeter of $\triangle DEJ$ to the length $DE$ and express this ratio in its simplest form.

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