
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 the functions $f(x) = \frac{6}{x}$ and $g(x) = x^2 + 1$,  
(1) Find the expression for $f[g(x)]$;  
(2) The solution set of the inequality $f[g(x)] \geq k - 7x^2$ is all real numbers, find the range of the real number $k$;  
(3) The solution set of the inequality $f[g(x)] > \frac{a}{x}$ contains exactly three positive integer solutions, find the range of the real number $a$.

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