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

(1) Find the value of $k$.

(2) If $f(1) < 0$, determine the monotonicity of $y=f(x)$ (no proof required) and find the range of values of $t$ for which the inequality $f(x^2 + tx) + f(4-x) < 0$ always holds.

(3) If $f(1) = \frac{3}{2}$, and $g(x) = a^{2x} + a^{-2x} - 2f(x)$, 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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