
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
Given a triangle \(ABC\). An incircle is inscribed in it, touching the sides \(AB\), \(AC\), and \(BC\) at the points \(C_1\), \(B_1\), \(A_1\) respectively. Find the radius of the excircle \(w\), which touches the side \(AB\) at the point \(D\), the extension of the side \(BC\) at the point \(E\), and the extension of the side \(AC\) at the point \(G\). It is known that \(CE = 6\), the radius of the incircle is 1, and \(CB_1 = 1\).

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