
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
A coordinate system is established with the origin as the pole and the positive half of the x-axis as the polar axis. Given the curve $C_1: (x-2)^2 + y^2 = 4$, point A has polar coordinates $(3\sqrt{2}, \frac{\pi}{4})$, and the polar coordinate equation of line $l$ is $\rho \cos (\theta - \frac{\pi}{4}) = a$, with point A on line $l$.

(1) Find the polar coordinate equation of curve $C_1$ and the rectangular coordinate equation of line $l$.

(2) After line $l$ is moved 6 units to the left to obtain $l'$, the intersection points of $l'$ and $C_1$ are M and N. Find the polar coordinate equation of $l'$ and the length of $|MN|$.

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