
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
Given triangle $ABC$, the sides are $AB = 13$, $BC = 30$, and $CA = 23$. The bisector of $\angle BAC$ intersects $\overline{BC}$ at point $D$ and the circumcircle of $\triangle ABC$ at point $E \ne A$. The circumcircle of $\triangle BED$ intersects the line $AB$ at points $B$ and $F \ne B$. Find the length of segment $CF$.

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