
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 $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 (a > b > 0)$ with eccentricity $e = \dfrac{\sqrt{3}}{2}$, $A$, $B$ are the left and right vertices of the ellipse, and $P$ is a point on the ellipse different from $A$ and $B$. The inclination angles of lines $PA$ and $PB$ are $\alpha$ and $\beta$ respectively. Then the minimum value of $|\tan \alpha - \tan \beta|$ is.

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