
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 right $\triangle DEF$ with legs $DE=5$, and $EF=12$. Find the length of the shorter angle trisector from $F$ to the hypotenuse.
A) $\frac{35\sqrt{3} - 60}{50}$
B) $\frac{1440\sqrt{3} - 600}{407}$
C) $\frac{720\sqrt{5} - 360}{203}$
D) $\frac{840\sqrt{11} - 480}{407}$

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