
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 that passes through point P (-1, 0) at an angle of $\alpha$ intersects the ellipse $\frac{x^2}{3} + \frac{y^2}{2} = 1$ at points M and N. 
(1) Write the parametric equation of line MN; 
(2) Find the minimum value of the product of the lengths PM and PN.

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