
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 function $f(x)=ax+b$, where $x\in[-1,1]$ and $a$, $b\in\mathbb{R}$ are constants.

$(1)$ If $a$ is chosen from the five numbers $-2$, $-1$, $0$, $1$, $2$, and $b$ is chosen from the three numbers $0$, $1$, $2$, find the probability that the function $y=f(x)$ is an odd function.

$(2)$ If $a$ is chosen from the interval $[-2,2]$ and $b$ is chosen from the interval $[0,2]$, find the probability that the function $y=f(x)$ has a zero.

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