
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 an initial velocity of $u$ at an angle of $\phi$ on various planets with different values of gravitational acceleration. The angles vary over $30^\circ \le \phi \le 150^\circ.$ For each planet, its trajectory's highest points when plotted form a closed curve. The area of this closed curve can be expressed as 
\[d \cdot \frac{u^4}{k^2}, \]
for each planet's specific gravity `k`. Find the value of `d` assuming gravity changes affect the height and the spread of the arches.

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