
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 a line $L$ passing through point $P\left(\frac{1}{2}, 1\right)$ with an inclination angle of $\alpha= \frac{\pi}{6}$, and in the polar coordinate system, the polar equation of circle $C$ is $\rho= \sqrt{2}\cos\left(\theta- \frac{\pi}{4}\right)$.  
(1) Write the parametric equation of line $L$ and convert the equation of circle $C$ into a Cartesian coordinate equation;  
(2) Suppose $L$ intersects circle $C$ at points $A$ and $B$, find the product of the distances from point $P$ to points $A$ and $B$.

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