
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 vertices $A, B, C$ respectively. Assume for simplicity that $H$ is inside the triangle $ABC$, which implies that all angles of the triangle are acute (such a triangle is called acute-angled). Determine the angles of the triangles $AH_{B}H_{C}, H_{A}BH_{C}, H_{A}H_{B}C$, 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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