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

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