
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 $f(x) = \frac{x^2}{ax+b}$ (where $a$ and $b$ are constants), and the equation $f(x) - x + 12 = 0$ has two real roots, 3 and 4.  
(1) Find the expression for $f(x)$.  
(2) If the two roots of $f(x) = -2m$ are $x_1$ and $x_2$, find the range of values for $x_1^2 + x_2^2$.  
(3) Solve the inequality $f(x) \geq \frac{1}{2-x}$.

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