
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 triangle $ABC$, and $H_A$, $H_B$, $H_C$ the feet of the altitudes from $A$, $B$, and $C$ respectively. Assume for simplicity that $H$ lies inside triangle $ABC$, which means all the angles of the triangle are acute (such a triangle is called acute-angled). 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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