
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 \( A_{1} A_{2} A_{3} \) be a triangle and \( \omega_{1} \) be a circle passing through \( A_{1} \) and \( A_{2} \). Assume that there are circles \( \omega_{2}, \ldots, \omega_{7} \) with the following properties:

(a) \( \omega_{k} \) passes through the points \( A_{k} \) and \( A_{k+1} \) for \( k=2,3, \ldots, 7, \quad (A_{i}=A_{i+3}) \)

(b) \( \omega_{k} \) and \( \omega_{k+1} \) touch each other externally for \( k=1,2, \ldots, 6 \).

Show: \( \omega_{1} = \omega_{7} \).

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