
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
Let the domain of the function $f_A(x)$ be $A=[a,b)$, and $f_A(x)=\left( \frac{x}{a} + \frac{b}{x} -1 \right)^2 - \frac{2b}{a} + 1$, where $a$ and $b$ are arbitrary positive real numbers, and $a < b$.
1. Find the minimum and maximum values of the function $f_A(x)$;
2. If $x_1 \in I_k = [k^2, (k+1)^2)$ and $x_2 \in I_{k+1} = [(k+1)^2, (k+2)^2)$, where $k$ is a positive integer, for all positive integers $k$, the inequality $f_{I_k}(x_1) + f_{I_{k+1}}(x_2) < m$ has a solution, find the range of $m$;
3. If for any $x_1, x_2, x_3 \in A$, $\sqrt{f_A(x_1)}, \sqrt{f_A(x_2)}, \sqrt{f_A(x_3)}$ can form the sides of a triangle, find the range of $\frac{b}{a}$.

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