
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 $g(x) \neq 0$. If $f'(x)g(x) - f(x)g'(x) > 0$, and $f(x) = a^{x}g(x)$ ($a > 0$ and $a \neq 1$) and $\frac{f(1)}{g(1)} + \frac{f(-1)}{g(-1)} = \frac{10}{3}$, then the value of $a$ is \_\_\_\_\_\_\_\_.

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