
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 $\phi$ from the horizontal. The trajectory of the projectile is given by the parametric equations:
\[
x = ut \cos \phi,
\]
\[
y = ut \sin \phi - \frac{1}{2} gt^2,
\]
where $t$ is time and $g$ is the acceleration due to gravity. Suppose $u$ is constant but $\phi$ varies from $0^\circ$ to $180^\circ$. As $\phi$ changes, the highest points of the trajectories trace a closed curve. The area enclosed by this curve can be expressed as $d \cdot \frac{u^4}{g^2}$. Find the value of $d$.

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