
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 an initial velocity of $u$ at an angle of $\theta$ from the ground, considering an air resistance that uniformly reduces the effective projectile velocity by a factor $k$, where $0 < k < 1$. The parametric equations modeling the trajectory of the projectile are:
\begin{align*}
x &= kut \cos \theta, \\
y &= kut \sin \theta - \frac{1}{2} gt^2.
\end{align*}
Find the area of the closed curve traced by the highest points of the projectile as $\theta$ varies from $0^\circ$ to $180^\circ$. Express the area in terms of $k$, $u$, and $g$.

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