
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 satisfy $f(x)=a^{x}\cdot g(x)$ where $a > 0$ and $a \neq 1$, and $f(x)\cdot g'(x) > f'(x)\cdot g(x)$ with $g(x) \neq 0$, and $\frac{f(1)}{g(1)}+ \frac{f(-1)}{g(-1)}= \frac{5}{2}$, find the value of $a$.

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