
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
Throw a uniform cube toy, which has numbers on its six faces: two faces with the number $0$, two faces with the number $2$, and two faces with the number $4$. Toss this toy twice in succession, using the numbers on the upward faces as the coordinates of point $P$ (the first toss as the x-coordinate and the second toss as the y-coordinate).

$(1)$ Calculate the probability that point $P$ falls within region $C$: $x^2+y^2\leqslant 10$;

$(2)$ If the maximum area polygon $M$ is formed with all the points falling within region $C$ as vertices, and a bean is randomly thrown onto region $C$, calculate the probability that the bean falls within the area of polygon $M$.

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