
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 passing through point $P\left( \frac{\sqrt{10}}{2},0\right)$ with an inclination angle of $\alpha$ intersects the curve $x^{2}+2y^{2}=1$ at points $M$ and $N$.  
$(1)$ Write a parametric equation for the line;  
$(2)$ Find the minimum value of $|PM|\cdot|PN|$ and the corresponding value of $\alpha$.

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