
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 $H_A, H_B, H_C$ be the feet of the altitudes from $A, B,$ and $C$. Assume for simplicity that $H$ is inside triangle $ABC$, which implies that all the angles of the triangle are acute (such a triangle is called an acute triangle). Determine the angles of 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 as $\widehat{A}, \widehat{B}$, and $\widehat{C}$.

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