
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 right triangle \(ABC\) with an acute angle of \(30^\circ\), an altitude \(CD\) is drawn from the right angle vertex \(C\). Find the distance between the centers of the inscribed circles of triangles \(ACD\) and \(BCD\), if the shorter leg of triangle \(ABC\) is 1.

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