
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 the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$ with an eccentricity of $e = \frac{\sqrt{3}}{2}$, let $A$ and $B$ be the left and right vertices of the ellipse, respectively, and $P$ be a point on the ellipse different from $A$ and $B$. If the slopes of lines $PA$ and $PB$ are denoted by $\alpha$ and $\beta$, respectively, find the minimum value of $|\tan \alpha - \tan \beta|$.

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