
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 $f(x) = ka^x - a^{-x}$ ($a > 0$ and $a \neq 1$) be an odd function defined on the domain $\mathbb{R}$.

(1) Find the value of $k$;  
(2) If $f(1) > 0$, find the solution set of the inequality $f(x^2 + 2x) + f(x - 4) > 0$;  
(3) If $f(1) = \frac{3}{2}$, and $g(x) = a^{2x} + a^{-2x} - 4f(x)$, find the minimum value of $g(x)$ on the interval $[1, +\infty)$.

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