
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 a linear function $f(x)$ is an increasing function on the set of real numbers $R$, and $f(f(x)) = 16x + 5$.  
$(1)$ Find the analytical expression of $f(x)$;  
$(2)$ If $g(x) = f(x)(x+m)$ and $g(x)$ is monotonically increasing on $(1, +\infty)$, find the range of real numbers for $m$.

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