
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
Given that the length of the line segment \( |AB| = 2a \), point \( P \) forms a right angle with line \( AB \), point \( Q \) is equidistant from \( A \) and \( B \), and \( \triangle QAB \) and \( \triangle PAB \) have equal areas (but are not zero). Point \( O \) is the midpoint of \( AB \), and \( OP \) intersects \( BQ \) at \( M \). Establish a coordinate system using \( O \) as the origin and the perpendicular bisector of \( AB \) as the \( y \)-axis. Let \( \angle BOP = \theta \). Determine the parametric equations of the trajectories of points \( P \), \( Q \), and \( M \) in terms of \( \theta \), convert these parametric equations into their standard form, and identify what type of curves these three trajectories are.

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