
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 $f(x) = a^x + (k-1)a^{-x} + k^2$ ($a>0$, $a \neq 1$) be an odd function defined on the domain $\mathbb{R}$.  
(1) Find the value of the real number $k$;  
(2) When $f(1) > 0$, find the range of values of the real number $t$ for which the inequality $f(x^2+x) + f(t-2x) > 0$ always holds;  
(3) If $f(1) = \frac{3}{2}$, let the function $g(x) = a^{2x} + a^{-2x} - 2mf(x)$, and if $g(x)$ has a minimum value of $-1$ on the interval $[1, +\infty)$, find the value of the real number $m$.

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