
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 fired with an initial velocity of $v$ at an angle of $\theta$ from the ground, and additionally, there is a constant wind resistance acting against the projectile in the vertical direction. This wind resistance can be modeled as $-kt$, where $k$ is a constant resistance factor, and $t$ denotes time. The modified equations of motion are:
\begin{align*}
x &= v t \cos(\theta), \\
y &= v t \sin(\theta) - \frac{1}{2} g t^2 - k t.
\end{align*}
The highest points of each trajectory are plotted as $\theta$ varies from $0^\circ$ to $180^\circ$. Determine the area enclosed by these points if expressed in the form $c \cdot \frac{v^4}{(g+k)^2}$.

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