
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
In the Cartesian coordinate system xOy, the parametric equation of $C_1$ is $$\begin{cases} x=1- \frac { \sqrt {2}}{2}t \\ y=1+ \frac { \sqrt {2}}{2}t \end{cases}$$ (where $t$ is the parameter). In the polar coordinate system with the origin as the pole and the positive x-axis as the polar axis, the polar equation of $C_2$ is $\rho^2-2\rho\cos\theta-3=0$.  
(Ⅰ) Convert the equation of $C_2$ into a Cartesian equation and identify the type of curve $C_2$ represents.  
(Ⅱ) $C_1$ and $C_2$ intersect at two points, A and B. Given a fixed point P with polar coordinates ($\sqrt {2}$, $\frac {\pi}{4}$), find the length of segment AB and the product of the distances from P to points A and B.

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