
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 that $f(x)$ and $g(x)$ are functions defined on $\mathbb{R}$ with $g(x) \neq 0$, $f'(x)g(x) > f(x)g'(x)$. Let $a$ and $b$ be the points obtained from two consecutive throws of the same die. If $f(x) - a^x g(x) = 0$ and $\frac{f(1)}{g(1)} + \frac{f(-1)}{g(-1)} \geq \frac{10}{3}$, calculate the probability that the equation $abx^2 + 8x + 1 = 0$ has two distinct real roots.

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