
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. Define positive real numbers $k_1$, $k_2$, and $k_3$, such that if the altitudes of triangle $ABC$ are extended past $A$, $B$, and $C$, to $A'$, $B'$, and $C'$, such that $AA' = k_1 BC$, $BB' = k_2 AC$, and $CC' = k_3 AB$, then triangle $A'B'C'$ is right-angled at $A'$. Find the conditions on $k_1$, $k_2$, and $k_3$.

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