
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 the $\triangle ABC$, let $AB=11$, $BC=13$, $CA=14$, and $\overline{AD}$ is an altitude. Points $E$ and $F$ lie on sides $\overline{AC}$ and $\overline{AB}$, respectively, so that $\overline{BE}$ and $\overline{CF}$ are angle bisectors, intersecting $\overline{AD}$ at $R$ and $S$, respectively. Find the length of $RS$.
A) $\frac{23\sqrt{95}}{171}$
B) $\frac{645\sqrt{5}}{4551}$
C) $\frac{645\sqrt{95}}{4551}$
D) $\frac{312\sqrt{95}}{351}$
E) $\frac{24\sqrt{95}}{13}$

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