
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 $DEF$ where hypotenuse $\overline{DE}=13$ and leg $\overline{DF}=5$, a bisector of angle $D$ meets side $\overline{EF}$ at point $D_1$. Another right triangle $XYZ$ is constructed such that hypotenuse $\overline{XY}=D_1E$ and leg $\overline{XZ}=D_1F$. Find the length of the bisector of angle $X$ that meets side $\overline{YZ}$ in $X_1$ if $\overline{D_1E} = \frac{3}{4} \overline{EF}$.
A) $\frac{6\sqrt{2}}{5}$
B) $\frac{3\sqrt{2}}{2}$
C) $\frac{9\sqrt{2}}{4}$
D) $\frac{12\sqrt{2}}{7}$
E) $\frac{15\sqrt{2}}{8}$

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