
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 $xoy$, with the origin $O$ as the pole and the positive half-axis of $x$ as the polar axis, the same unit length is used to establish the polar coordinate system. It is known that the curve $C_{1}: x^{2}+y^{2}=1$, and the line $l: \rho(\cos \theta-\sin \theta)=4$.  
$(1)$ After stretching the horizontal and vertical coordinates of all points on the curve $C_{1}$ by $2$ times and $\sqrt{3}$ times respectively, we get the curve $C_{2}$. Please write down the Cartesian equations of the line $l$ and the curve $C_{2}$.  
$(2)$ If the line $l_{1}$ passes through the point $P(1,2)$ and $l_{1} \parallel l$, and $l_{1}$ intersects the curve $C_{2}$ at points $M$ and $N$, find the value of $|PM|\cdot|PN|$.

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