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

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