
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

Determine the maximum possible value of the tangent of the angle at which pucks can be ejected from an automatic puck shooter. The trajectory of a puck, considered as a material point in its plane of flight in a coordinate system centered at point $O$, with the x-axis directed along the ice surface, is described by the equations:
\[
\begin{cases}
x = d + V_{0} t \cos \alpha, \\
y = V_{0} t \sin \alpha - \frac{g t^2}{2}.
\end{cases}
\]
where $d = 16 \, \text{m}$, $V_{0} = 20 \, \text{m/s}$, and the acceleration due to gravity $g = 10 \, \text{m/s}^2$. The goal is to ensure the puck trajectory does not exceed a certain height and intersects the goal plane on the descending branch of its trajectory. The goal line is 25 meters from the central point $O$ of the hockey rink.

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