
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 the functions $f(x)$ and $g(x)$ defined on $\mathbb{R}$ satisfy $\frac{f(x)}{g(x)} = a^x$ and $f'(x) g(x) < f(x) g'(x)$, and that $\frac{f(1)}{g(1)} + \frac{f(1)}{g'(1)} = \frac{5}{2}$, if the sum of the first $n$ terms of the sequence ${\frac{f(n)}{g(n)} \mid n \in \mathbb{N}^*}$ is $\frac{31}{32}$, then $n = \_\_\_\_\_\_.$

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