
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 with varying initial velocities from $v_0$ to $2v_0$ at a fixed angle of $45^\circ$ from the ground. The motion of the projectile can be modeled by the parametric equations
\[
x = vt \cos \theta, \quad y = vt \sin \theta - \frac{1}{2} gt^2,
\]
where $t$ denotes time, $g$ denotes acceleration due to gravity, and $\theta = 45^\circ$. As $v$ varies from $v_0$ to $2v_0$, the highest points of the trajectories will trace a curve. Determine the area of this curve.

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