
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 triangle $ABC$, sides $AC=5$, $BC=12$, and $AB=13$. Point $D$ is on $\overline{AB}$, and $\overline{CD}$ bisects the right angle at $C$. The circles inscribed in triangles $ADC$ and $BCD$ have radii $r_a$ and $r_b$ respectively. Find the ratio $r_a/r_b$.
A) $\frac{10 - \sqrt{2}}{30}$
B) $\frac{13 - \sqrt{2}}{28}$
C) $\frac{8 - \sqrt{2}}{26}$
D) $\frac{15 - \sqrt{2}}{32}$
E) $\frac{11 - \sqrt{2}}{24}$

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