
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
Let \( \triangle ABC \) have side lengths \(a, b, c\) with \( b > \max \{a, c\} \). Given three nonzero real numbers \( x_0, y_0, z_0 \) that satisfy the line \( ax + by + c = 0 \) passing through the point \( \left(\frac{z_0}{x_0}, \frac{2 y_0}{x_0}\right) \), and the point \( \left(\frac{z_0}{y_0}, \frac{x_0}{y_0}\right) \) lying on the ellipse \( x^2 + \frac{y^2}{4} = 1 \), find the maximum value of \( \tan B \cdot \cot C \).

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