
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 $\phi$ from the ground. The trajectory is modeled by:
\begin{align*}
x &= ut \cos \phi, \\
y &= ut \sin \phi - \frac{1}{2} gt^2,
\end{align*}
where $t$ is time and $g$ is the acceleration due to gravity. Suppose $u$ is constant, but $\phi$ varies over $0^\circ \le \phi \le 90^\circ$. The highest point of each trajectory is plotted. As $\phi$ varies, the highest points of the trajectories trace a closed curve. The area of this closed curve can be expressed in the form
\[d \cdot \frac{u^4}{g^2}.\]
Find $d$.

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