
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 function $f(x) = \ln|x|$ ($x \neq 0$), and the function $g(x) = \frac{1}{f'(x)} + af'(x)$ ($x \neq 0$)  
(1) When $x \neq 0$, find the expression for the function $y = g(x)$;  
(2) If $a > 0$, and the minimum value of the function $y = g(x)$ on $(0, +\infty)$ is 2, find the value of $a$;  
(3) Under the condition of (2), find the area of the shape enclosed by the line $y = \frac{2}{3}x + \frac{7}{6}$ and the graph of the function $y = g(x)$.

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