
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 the following conditions:
(1) $f(x) = 2a^xg(x)$, where $a > 0$ and $a \neq 1$;
(2) $g(x) \neq 0$;
(3) $f(x)g'(x) < f'(x)g(x)$ and $\frac{f(1)}{g(1)} + \frac{f(-1)}{g(-1)} = 5$, determine the value of $a$.

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