
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
The first circle centered at point \( O \) is inscribed in triangle \( ABC \). Points \( A \) and \( B \) lie on a second circle with the same center \( O \). Line \( AC \) intersects the second circle at point \( D \) (\( D \neq A \)), and line \( BC \) intersects the second circle at point \( E \) (\( B \neq E \)). It is known that angle \( ABC \) is equal to angle \( CAE \). Find the cosine of angle \( BAC \).

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