
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
A line $l$ with a slope angle of $α$ passes through the point $P( \frac { \sqrt {10}}{2},0)$ and intersects the curve $x^{2}+12y^{2}=1$ at points $M$ and $N$. Find the minimum value of $|PM|⋅|PN|$ and the corresponding value of $\tan(α)$.

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