
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, given point $B(1,1)$, the parametric equation of curve $C$ is $\begin{cases} x=2\cos \theta \\ y= \sqrt {3}\sin \theta \end{cases}$ ($\theta$ is the parameter), with the origin as the pole and the positive $x$-axis as the polar axis, the polar coordinates of point $A$ are $(4 \sqrt {2}, \frac {\pi}{4})$, the polar equation of line $l$ is $\rho\cos (\theta- \frac {\pi}{4})=a$, and $l$ passes through point $A$. The line parallel to $l$ and passing through point $B$ is $l_{1}$, $l_{1}$ intersects curve $C$ at two points $M$, $N$  
(Ⅰ) Find the minimum distance from points on curve $C$ to line $l$  
(Ⅱ) Find the value of $|AB|$.

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