
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
Consider a projectile that is fired from an initial height $h$ above the ground with an initial velocity $v$ at an angle $\theta$ from the horizontal. The projectile's path follows the parametric equations given by:
\[
x = vt \cos \theta, \quad y = h + vt \sin \theta - \frac{1}{2} gt^2
\]
where $g$ is the acceleration due to gravity. Let $\theta$ vary over $0^\circ \le \theta \le 90^\circ$. Determine the area enclosed by the locus of the highest points of each trajectory if $h = \frac{v^2}{8g}$.

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