
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 points \( A_{0} \), \( B_{1} \), and \( C_{2} \). Construct a triangle \( ABC \) such that:
- The foot of the altitude from \( A \) intersects the opposite side at \( A_{0} \).
- The midpoint of the side \( AC \) is at \( B_{1} \).
- The midpoint of the altitude from \( C \) intersects the altitude at \( C_{2} \).

Determine the number of possible solutions based on the relative positions of the given points.

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