
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
In an acute-angled triangle \( ABC \), the altitude \( CE \) on \( AB \) and the altitude \( BD \) on \( AC \) intersect at point \( H \). A circle with diameter \( DE \) intersects \( AB \) and \( AC \) at points \( F \) and \( G \) respectively. Line \( FG \) intersects \( AH \) at point \( K \). Given that \( BC = 25 \), \( BD = 20 \), and \( BE = 7 \), find the length of \( AK \).

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