
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
Two dice are rolled simultaneously, and the numbers facing up are recorded as $a$ and $b$, respectively. Consider the function $f(x)=(a-b)x^{2}+bx+1$.
(1) Find the probability that $f(x)$ is an even function.
(2) Find the probability that $f(x)$ is strictly increasing on $\left[- \frac{1}{2},+\infty\right)$.

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