
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 a triangle. There exists a positive real number $k$, such that if the medians of triangle $ABC$ are extended past $A$, $B$, and $C$, to $A'$, $B'$, and $C'$, such that $AA' = k BC$, $BB' = k AC$, and $CC' = k AB$, then triangle $A'B'C'$ is equilateral.

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