
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
The inscribed circle of the right triangle \(ABC\) (with the right angle at \(C\)) touches the sides \(AB\), \(BC\), and \(CA\) at the points \(C_1\), \(A_1\), and \(B_1\) respectively. The altitudes of the triangle \(A_1B_1C_1\) intersect at the point \(D\). Find the distance between the points \(C\) and \(D\), given that the lengths of the legs of the triangle \(ABC\) are 3 and 4.

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