
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
Two circles with the same radius of 5 intersect at points \( A \) and \( B \). Point \( C \) is selected on the first circle, and point \( D \) is selected on the second circle. It turns out that point \( B \) lies on the segment \( C D \), and \(\angle C A D=90^\circ\). On the perpendicular to \( C D \) passing through point \( B \), a point \( F \) is chosen such that \( B F = B D\) (points \( A \) and \( F \) are located on opposite sides of the line \( C D \)). Find the length of the segment \( C F \).

Additionally, suppose that \( B C = 6 \). Find the area of triangle \( A C F \).

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