
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 varying initial velocities $v$, but at a constant angle $\theta = 45^\circ$ from the ground. The trajectory of the projectile could again be modeled by the parametric equations:
\begin{align*}
x &= vt \cos(45^\circ), \\
y &= vt \sin(45^\circ) - \frac{1}{2} gt^2,
\end{align*}
where $t$ denotes time and $g$ denotes the acceleration due to gravity. As $v$ changes, plot the highest point of each trajectory and determine the equation of the curve that these points trace.

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