
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 the angles of triangle \(ABC\) be \(\angle CAB = \alpha\) and \(\angle CBA = \beta\). Then,

\[ 
\angle B'AO = \angle CAO = 90^\circ - \beta \quad \text{and} \quad \angle A'BO = \angle CBO = 90^\circ - \alpha.
\]

Therefore, the distance from point \(A'\) to the line \(BO\) is equal to \(A'B \sin \angle A'BO = A'B \sin (90^\circ - \alpha) = A'B \cos \alpha = AB \cos \beta \cos \alpha\), and the distance from point \(B'\) to the line \(AO\) is equal to \(AB' \sin \angle B'AO = AB' \sin (90^\circ - \beta) = AB' \cos \beta = AB \cos \alpha \cos \beta\).

Thus, these distances are equal.

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