
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
In the given triangle \( OAB \), the angle \( AOB \) is less than \( 90^\circ \). From any arbitrary point \( M \) (other than \( O \)) inside the perimeter or interior of \( \triangle AOB \), perpendiculars are drawn to the lines \( \overline{OA} \) and \( \overline{OB} \). Let the feet of these perpendiculars be denoted by \( P \) and \( Q \) respectively. Furthermore, let \( H \) be the orthocenter of triangle \( OPQ \).

What is the locus of the points \( H \) if \( M \) traverses

a) the side \( AB \);

b) the interior of triangle \( OAB \)?

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