
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 launched with an initial velocity of $u$ at an angle of $\alpha$ from the ground. The trajectory can be modeled by the parametric equations:
\[
x = ut \cos \alpha, \quad y = ut \sin \alpha - \frac{1}{2} kt^2,
\]
where $t$ denotes time and $k$ denotes a constant acceleration, forming a parabolic arch.

Suppose $u$ is constant, but $\alpha$ varies over $0^\circ \le \alpha \le 90^\circ$. The highest points of each parabolic arch are plotted. Determine the area enclosed by the curve traced by these highest points, and express it in the form:
\[
d \cdot \frac{u^4}{k^2}.
\]

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