
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
A projectile is fired from ground level with an initial velocity $v$ and at an angle $\theta$ to the horizontal, where $0^\circ \leq \theta \leq 90^\circ$. The motion of the projectile is modeled by the parametric equations:
\[
x = vt \cos \theta, \quad y = vt \sin \theta - \frac{1}{2} gt^2,
\]
where $t$ represents time, and $g$ is the acceleration due to gravity. As $\theta$ varies within the given range, the highest points of the trajectories trace a curve. Determine the area enclosed by this curve.

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