
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
Since it is required that the stone does not touch the specified segment, the desired speed interval has the form \(v > v_{0}\). Let's call \(v_{0}\) the minimum speed.

By choosing the throwing position and the angle of inclination of the initial speed to the horizon, the trajectory can be made to pass through any two given points that do not lie on the same vertical line. From the law of conservation of energy, it follows that the case of the minimum speed will correspond to the trajectory passing through both corners of the building.

Let the stone, when passing through the upper part of the building, have a speed \(v_{1}\), directed at an angle \(\beta\) to the horizon. The geometric locus of the points through which all possible trajectories pass is described by the inequality

\[ y \leq \frac{v_{1}}{2 g}-\frac{g x^{2}}{2 v_{1}^{2}} \]

in a coordinate system with the origin at the top of the building and the \(x\) and \(y\) axes directed horizontally towards the slope and vertically (safety parabola).

The smallest value of \(v_{1}\) corresponds to the case

\[ -l \operatorname{tg} \alpha = \frac{v_{1}}{2 g}-\frac{g l^{2}}{2 v_{1}^{2}} \]

In this case, the trajectory passes through the two vertices of the building and due to the convexity of the parabola does not intersect the roof. Hence,

\[ v_{1}^{2} = g l \left(\sqrt{\operatorname{tg}^{2} \alpha + 1} - \operatorname{tg} \alpha \right) = g l \frac{1 - \sin \alpha}{\cos \alpha} \]

According to the law of conservation of energy, the minimum speed of the stone at ground level is

\[ v_{0} = \sqrt{g \left(2 H + l \frac{1 - \sin \alpha}{\cos \alpha} \right)} \]

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