
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 a height \( h \) above the ground with an initial velocity \( v \) at an angle \( \theta \) with the horizontal. The projectile's motion can be described by the parametric equations
\[
x = vt \cos(\theta),
\]
\[
y = h + vt \sin(\theta) - \frac{1}{2} gt^2,
\]
where \( t \) is time and \( g \) is the acceleration due to gravity. The angle \( \theta \) varies from \( 0^\circ \) to \( 90^\circ \). As \( \theta \) varies, plot the highest points of each arch. Determine the shape of the curve traced by these highest points and calculate the area enclosed by this curve, expressed in terms of \( v, g, \) and \( h \).

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