
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 $g(x) \neq 0$, it is known that $f(x)g'(x) > f'(x)g(x)$. If $f(x) = a^x g(x)$ where $a > 0$ and $a \neq 1$, and $$\frac{f(1)}{g(1)} + \frac{f(-1)}{g(-1)} = \frac{5}{2},$$ in the finite sequence $\left\{ \frac{f(n)}{g(n)} \right\}$ (where $n=1,2,\ldots,10$), calculate the probability that the sum of the first $k$ terms is greater than $\frac{15}{16}$ for an arbitrary positive integer $k$ with $1 \leq k \leq 10$.

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