
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 the ground with an initial velocity of $v$ at an angle of $\theta$ to the horizontal and an additional vertical velocity $u$. The trajectory can then be modeled by the parametric equations:
\begin{align*}
x &= vt \cos \theta, \\
y &= (vt \sin \theta + ut) - \frac{1}{2} gt^2.
\end{align*}
As $\theta$ varies over $0^\circ \leq \theta \leq 180^\circ$, the highest points of the arches trace a closed curve. The area of this closed curve can be expressed in the form $c \cdot \frac{(v^2 + u^2)^2}{g^2}$. Find $c$.

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