
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
Elective 4-4: Coordinate System and Parametric Equations

Given that the polar equation of line $l$ is $\rho \sin (\theta -\frac{\pi }{4})=\frac{\sqrt{2}}{2}$, and the parametric equation of circle $C$ is:
$$
\begin{cases} 
& x=2\cos \theta \\
& y=-2+2\sin \theta 
\end{cases}
$$
where $\theta$ is the parameter.

(Ⅰ) Determine the positional relationship between line $l$ and circle $C$.

(Ⅱ) If the parametric equation of the ellipse is:
$$
\begin{cases} 
& x=3\cos \phi \\
& y=\sqrt{3}\sin \phi 
\end{cases}
$$
where $\phi$ is the parameter, and a line $l'$ passes through the center of circle $C$ and is perpendicular to line $l$, intersecting the ellipse at points $A$ and $B$, find $|AB|$.

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