
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 \( H \) be the orthocenter of a triangle \( ABC \), and let \( H_A, H_B, H_C \) be the feet of the altitudes from \( A, B, \) and \( C \), respectively. To simplify, assume that \( H \) is inside the triangle \( ABC \), which implies that all angles of the triangle are acute (such a triangle is called an acute triangle). Determine the angles of the triangles \( A H_B H_C \), \( H_A B H_C \), \( H_A H_B C \), and \( H_A H_B H_C \), in terms of the angles of triangle \( ABC \), denoted by \(\widehat{A}, \widehat{B},\) and \(\widehat{C}\).

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