
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
Suppose a projectile is launched from the ground with different initial velocities ranging from $u$ to $2u$ at a fixed angle of $45^\circ$. The motion of the projectile is described by the parametric equations
\begin{align*}
x &= v t \cos(45^\circ), \\
y &= v t \sin(45^\circ) - \frac{1}{2} g t^2,
\end{align*}
where $v$ varies from $u$ to $2u$, $t$ is the time, and $g$ is the acceleration due to gravity. Determine the area of the region traced by the highest points of the projectile trajectories as $v$ varies.

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