
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 that triangle $ABC$ has $AC=3$, $BC=4$, and $AB=5$, with point $D$ on $\overline{AB}$ and $\overline{CD}$ bisecting the right angle, find the ratio of the radii of the inscribed circles of $\triangle ADC$ and $\triangle BCD$.

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