
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 set $A=\{-2,0,1,3\}$, in the Cartesian coordinate system, the coordinates of point $M$ are $(x,y)$, satisfying $x\in A$ and $y\in A$.

$(1)$ Please list all the coordinates of point $M$;

$(2)$ Calculate the probability that point $M$ is not on the $y$-axis;

$(3)$ Calculate the probability that point $M$ falls exactly within the region $\begin{cases} x+y-5 < 0, \\ x > 0, \\ y > 0 \end{cases}$.

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