
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 $g(x)\neq 0$, and $f'(x)g(x) < f(x)g'(x)$, and $f(x) = a^x g(x)$, and $\frac{f(1)}{g(1)} + \frac{f(-1)}{g(-1)} = \frac{5}{2}$, determine the probability that the sum of the first $k$ terms of the sequence $\left\{\frac{f(n)}{g(n)}\right\}$ for $n=1,2,\ldots,10$ is greater than $\frac{15}{16}$.

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