
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 of \( 2v \) at an angle \( \theta \) relative to the horizontal. Its trajectory is modeled by the parametric equations:
\[
x = 2vt \cos \theta, \quad y = 2vt \sin \theta - \frac{1}{2} gt^2 + h,
\]
where \( t \) is the time, \( g \) is the acceleration due to gravity, and \( v \) is a constant speed. Suppose \( h \) is held constant, but \( \theta \) is allowed to vary from \( 0^\circ \) to \( 180^\circ \). As \( \theta \) varies, the highest points of the arches trace a closed curve. The area enclosed by this curve can be expressed in the form \( c \cdot \frac{4v^4}{g^2} \). Determine \( c \).

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