
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
The velocity of the free fall of a ball is given by the law of conservation of energy $V=\sqrt{2 g r \cos \alpha}$. The free fall of the ball is described by the following equations:

$$
x(t)=R \sin \alpha+V \cos \alpha t, \quad y(t)=R(1-\cos \alpha)+V \sin \alpha t-\frac{g t^{2}}{2}
$$

The flight time $T$ is found from the quadratic equation $y(T)=0$:

$$
T=\sqrt{\frac{2 R}{g}}\left[\sin \alpha \sqrt{\cos \alpha}+\sqrt{1-\cos ^{3} \alpha}\right]
$$

From this, we find the desired coordinate of the point of impact:

$$
x(T)=R\left[\sin \alpha+\sin 2 \alpha+\sqrt{\cos \alpha\left(1-\cos ^{3} \alpha\right)}\right]
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Present the answer in LaTex format: \boxed{Your answer}
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