
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
Let  $ABC$  be at triangle with incircle  $\Gamma$ . Let  $\Gamma_1$ ,  $\Gamma_2$ ,  $\Gamma_3$  be three circles inside  $\triangle ABC$  each of which is tangent to  $\Gamma$  and two sides of the triangle and their radii are  $1,4,9$ . Find the radius of  $\Gamma$ .

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