
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
Consider the triangle \(ABC\) with the property that none of the sides of the triangle are greater than \(BC\) nor smaller than \(AB\). Denote the feet of the altitudes from vertices \(A\), \(B\), and \(C\) as \(A_1\), \(B_1\), and \(C_1\) respectively, and the midpoints of sides \(BC\), \(CA\), and \(AB\) as \(A_2\), \(B_2\), and \(C_2\) respectively. Traverse the path \(A, C_1, B, A_1, C, B_1, A\) and determine where the points \(A_2\), \(B_2\), and \(C_2\) lie between which two points along this path for various shapes of the triangle, and describe the possible backtracks and coincidences between the points along the path. How does the orientation of triangles \(A_1B_1C_1\) and \(A_2B_2C_2\) compare to the orientation of triangle \(ABC\)? Express the angles between the line pairs \(B_1C_2\) and \(C_1B_2\), \(C_1A_2\) and \(A_1C_2\), and \(A_1B_2\) and \(B_1A_2\) in terms of the angles of the triangle.

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