
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 $xOy$, with $O$ as the origin and the positive half-axis of $x$ as the basis for establishing a polar coordinate system, the polar equation of curve $C$ is $\rho^{2}-4\rho\sin \theta+3=0$, and the parametric equation of line $l$ is $\begin{cases} x=1- \dfrac { \sqrt {2}}{2}t \\ y=3+ \dfrac { \sqrt {2}}{2}t\end{cases}$, ($t$ is the parameter).

$(1)$ Write the Cartesian coordinate equations for curve $C$ and line $l$;  
$(2)$ If points $A$ and $B$ are two moving points on curve $C$, and point $P$ is a moving point on line $l$, find the maximum value of $\angle APB$.

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