
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
Consider  $\triangle \natural\flat\sharp$ .  Let  $\flat\sharp$ ,  $\sharp\natural$  and  $\natural\flat$  be the answers to problems  $4$ ,  $5$ , and  $6$ , respectively.  If the incircle of  $\triangle \natural\flat\sharp$  touches  $\natural\flat$  at  $\odot$ , find  $\flat\odot$ .

*Proposed by Evan Chen*

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