
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 with an initial velocity of $u$ from the surface of different celestial bodies, each with its own gravitational constant $g_k$, at a fixed angle of $45^\circ$. The projectile's trajectory from each launch site can be modeled by the equations:
\begin{align*}
x_k &= u t \cos 45^\circ, \\
y_k &= u t \sin 45^\circ - \frac{1}{2} g_k t^2,
\end{align*}
for $k = 1, 2, \ldots, n$. Define the highest point of each trajectory for different constants $g_k$. Plot these points and determine the area of the enclosed curve, expressed as a function relative to $u$ and averaged over these gravitational constants, $G = \frac{1}{n}\sum_{k=1}^{n} g_k$.

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