
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
Given a particle projected with an initial velocity of $180$ feet per second from the ground at an angle of $45^\circ$ to the horizontal, derive the expressions for the horizontal range $R$ and the maximum height $H$ by using the given expression $s = 180t \sin(45^\circ) - 16t^2$ where $s$ is the vertical displacement in feet as a function of time $t$ in seconds.

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