
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 the acute triangle $ABC$, the altitude $CE$ from $C$ onto $AB$ intersects the altitude $BD$ from $B$ onto $AC$ 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}
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