
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 $C$: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = l (a > b > 0)$, with $F_{1}$ and $F_{2}$ as the left and right foci, and $B_{1}$ as the lower vertex. The line $l$ passing through $F$ intersects the ellipse at points $M$ and $N$. When the angle of inclination of line $l$ is $\frac{\pi}{6}$, $F_{1}B \perp l$.
(I) Find the eccentricity of the ellipse $C$;
(II) If $P$ is a moving point on the ellipse, and the slopes of $PM$ and $PN$ are denoted by $k_{PM}$ and $k_{PN}$ respectively (and are non-zero), does $| \frac{1}{k_{PM}} - \frac{1}{k_{PN}} |$ have a minimum value when line $l$ is perpendicular to the $x$-axis? If so, find this minimum value; if not, explain the reason.

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