
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 that triangle $\triangle ABC$ has side lengths $a$, $b$, and $c$, where $b > \max \{a, c\}$, there are three nonzero real numbers $x_{0}$, $y_{0}$, and $z_{0}$ such that the line $a x + b y + c = 0$ passes through the point $\left(\frac{z_{0}}{x_{0}}, \frac{2 y_{0}}{x_{0}}\right)$. If the point $\left(\frac{z_{0}}{y_{0}}, \frac{x_{0}}{y_{0}}\right)$ lies on the ellipse $x^{2} + \frac{y^{2}}{4} = 1$, then determine the maximum value of $\tan B \cdot \cot C$.

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