
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 $u$ at an angle $\phi$ from the ground. The trajectory can be described by the parametric equations:
\[
x = ut \cos \phi, \quad y = ut \sin \phi - \frac{1}{2} g t^2,
\]
where $t$ is the time, $g$ is the acceleration due to gravity, and the angle $\phi$ varies over $0^\circ \le \phi \le 90^\circ$. The highest points of each trajectory are plotted as $\phi$ changes. Determine the area of the curve traced by these highest points, expressed as $d \cdot \frac{u^4}{g^2}$.

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