
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 rectangular coordinate system $(xoy)$, a polar coordinate system is established with the coordinate origin as the pole and the non-negative semi-axis of the $x$-axis as the polar axis. The polar coordinate equation of circle $C$ is $\rho =4\cos \theta $.

(1) Find the rectangular coordinate equation of circle $C$;

(2) Given that circle $C$ intersects with the $x$-axis at points $A$ and $B$, if there exists a point $P$ on line $l$: $y=2x+2m$ such that $\angle APB={90}^{\circ }$, find the maximum value of the real number $m$.

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