
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
From a point, objects are thrown at equal speeds but at different angles. Determine the geometric locus of the vertices of the parabolic trajectories that are in the same vertical plane.

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