
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
Given positive numbers \( h, s_{1}, s_{2} \) and a triangle \( A B C \) in space, how many ways can one choose a point \( D \) such that in the tetrahedron \( A B C D \) the height dropped from the vertex \( D \) is equal to \( h \), and the areas of the faces \( A C D \) and \( B C D \) are \( s_{1} \) and \( s_{2} \) respectively? (Investigate all possible cases.)

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