
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 the point $P\left( \frac {1}{2}, 1\right)$ with an inclination angle of $\alpha= \frac {\pi}{6}$, and 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 Cartesian coordinates;  
(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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