
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, with the origin as the pole and the positive half-axis of $x$ 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 $\rho\cos(\theta- \frac{\pi}{4})=a$, and $l$ passes through point $A$. The parametric equation of curve $C_{1}$ is 
$$
\begin{cases} 
x=2\cos\theta \\ 
y=\sqrt{3}\sin\theta 
\end{cases}
(\theta \text{ is the parameter}).
$$

$(1)$ 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 $|BM|\cdot|BN|$.

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