
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 $xOy$, establish a polar coordinate system with the origin $O$ as the pole and the non-negative half-axis of the $x$-axis as the polar axis. The polar equation of circle $C$ is $\rho=4\cos \theta$.
$(1)$ Find the Cartesian coordinate equation of circle $C$;
$(2)$ Given that circle $C$ intersects the $x$-axis at points $A$ and $B$, the line $l: y=2x$ is symmetrical to line $l'$ about the point $M(0,m)$. If there exists a point $P$ on $l'$ such that $\angle APB=90^{\circ}$, find the maximum value of $m$.

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