
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 an equilateral triangle with side length 1. Let \( D \) be the reflection of point \( C \) over point \( A \). On the line \( AB \) beyond point \( B \), there is a point \( F \) such that \( EF=1 \), where \( E \) is the intersection point of the lines \( FC \) and \( DB \). Determine the lengths of the segments \( CE = x \) and \( BF = y \).

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