
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
Triangle $DEF$ has side lengths $DE = 13$, $EF = 30$, and $FD = 23$. The bisector of $\angle DEF$ intersects $\overline{EF}$ in point $G$, and intersects the circumcircle of $\triangle DEF$ in point $H \ne D$. The circumcircle of $\triangle DGH$ intersects the line $DE$ in points $D$ and $I \ne D$. Find the length of $FI$.

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