
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 of the same radius 9 intersect at points \(A\) and \(B\). A point \(C\) is chosen on the first circle, and a point \(D\) is chosen 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 \(B\), a point \(F\) is chosen such that \(BF = BD\) (points \(A\) and \(F\) are on opposite sides of the line \(CD\)). Find the length of the segment \(CF\).

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