
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. $f(x) = a^x \cdot g(x)$ ($a>0$, $a \neq 1$);
2. $g(x) \neq 0$;
3. $f(x) \cdot g'(x) > f'(x) \cdot g(x)$;

If $\frac{f(1)}{g(1)} + \frac{f(-1)}{g(-1)} = \frac{5}{2}$, then $a = \_\_\_\_\_\_$.

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