
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
(Selected Series 4: Coordinate System and Parametric Equations)

In the polar coordinate system established with the origin as the pole and the positive half-axis of _x_ as the polar axis, it is known that the curve $C1: (x-2)^{2}+y^{2}=4$, the polar coordinates of point $A$ are $(3\sqrt{2}, \frac{\pi}{4})$, and the polar equation of line _l_ is $\rho\cos (\theta- \frac{\pi}{4})=a$, and point $A$ is on line _l_.  
(1) Find the polar equation of curve $C_{1}$ and the Cartesian equation of line _l_;  
(2) Suppose _l_ is translated 6 units to the left to obtain _l_′, _l_′ intersects $C_{1}$ at points $M$, $N$, find the polar equation of _l_′ and the length of $|MN|$.

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