
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 from the ground with an initial speed $u$ at an angle of $\phi$. The motion of the projectile can be modeled by the parametric equations:
\[
x = u t \cos \phi, \quad y = u t \sin \phi - \frac{1}{2} g t^2,
\]
where $t$ denotes time, $g$ is the acceleration due to gravity, and $\phi$ varies from $0^\circ$ to $180^\circ$. Determine the area enclosed as the highest points of the trajectories form a closed curve, expressed in the form $k \cdot \frac{u^4}{g^2}$, and find the value of $k$.

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