
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
Let $ABC$ be an acute triangle. There exists a positive real number $k$, such that if the altitudes of triangle $ABC$ are extended past points $A$, $B$, and $C$, to points $A'$, $B'$, and $C'$ respectively, such that \(AA' = 2k \cdot BC\), \(BB' = 2k \cdot AC\), and \(CC' = 2k \cdot AB\), then the triangle $A'B'C'$ is equilateral. Determine the value of $k$.

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