
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
a) Two circles of the same radius 10 intersect at points \(A\) and \(B\). Point \(C\) is selected on the first circle, and point \(D\) on the second circle. It turns out that point \(B\) lies on the segment \(CD\) and \(\angle CAD = 90^{\circ}\). On the perpendicular to \(CD\) passing through point \(B\), point \(F\) is chosen such that \(BF = BD\) (points \(A\) and \(F\) are on opposite sides of line \(CD\)). Find the length of segment \(CF\).

b) Let it be additionally known that \(BC = 12\). Find the area of triangle \(ACF\).

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