
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
The side length of an equilateral triangle $ABC$ is 4. Point $D$ is the midpoint of side $BC$. A line passes through point $B$ and intersects side $AC$ at point $M$. Perpendiculars $DH$ and $AK$ are dropped from points $D$ and $A$ onto the line $BM$. Calculate the length of segment $AM$ if
$$
A K^{4} - D H^{4} = 15
$$

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