
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 \( ABC \) located in space. How many ways can one choose point \( D \) such that in the tetrahedron \( ABCD \), the height dropped from vertex \( D \) is equal to \( h \), and the areas of the faces \( ACD \) and \( BCD \) are \( s_{1} \) and \( s_{2} \) respectively (consider all possible cases)?

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