
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
Given the polar equation of curve $C_{1}$ is $\rho\cos (\theta- \frac {\pi}{3})=-1$, and the polar equation of curve $C_{2}$ is $\rho=2 \sqrt {2}\cos (\theta- \frac {\pi}{4}).$ Establish a plane rectangular coordinate system with the pole as the coordinate origin and the polar axis as the positive half-axis of $x$.  
$(1)$ Find the rectangular coordinate equation of curve $C_{2}$.  
$(2)$ Find the maximum distance from a moving point $M$ on curve $C_{2}$ to curve $C_{1}$.

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