
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
### Part ①
In the Cartesian coordinate system, with the origin as the pole and the positive half of the $x$-axis as the polar axis, a polar coordinate system is established. The polar coordinates of point $A$ are $(4\sqrt{2},\frac{\pi }{4})$, and the polar equation of line $l$ is $\rho \cos (\theta -\frac{\pi }{4})=a$, with $l$ passing through point $A$. The parametric equation of curve $C_{1}$ is $\begin{cases} & x=2\cos \theta, \\ & y=\sqrt{3}\sin \theta \end{cases}$ ($\theta$ is the parameter).

(1) Find the maximum and minimum distances from points on curve $C_{1}$ to line $l$;

(2) A line $l_{1}$, parallel to line $l$ and passing through point $B(-2,2)$, intersects curve $C_{1}$ at points $M$ and $N$. Find the value of $|BM|\cdot |BN|$.

### Part ②
Given the function $f(x)=|x+a|+|x-2|$

(1) When $a=-3$, find the solution set of the inequality $f(x)\geqslant 3$;

(2) If the solution set of $f(x)\leqslant |x-4|$ includes $[1,2]$, find the range of the real number $a$.

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