
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 $f(x) = -x^2 + ax - b$.

$(1)$ If $a$ and $b$ are each chosen from the numbers $0, 1, 2, 3, 4$, calculate the probability that the function has a root.

$(2)$ If $a$ and $b$ are each chosen from the interval $[0, 4]$, calculate the probability that $f(1) > 0$.

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