
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 the exponential function $y=g(x)$ satisfies: $g(2)=4$, and the function $f(x) = \frac{-g(x)+a}{2g(x)+b}$ defined on $\mathbb{R}$ is an odd function.  
(1) Find the values of $a$ and $b$;  
(2) Determine the monotonicity of the function $f(x)$ (conclusion only, no proof required);  
(3) If for any $t \in [0, 1]$, the inequality $f(t^2-2t) + f(2t^2-k) > 0$ always holds, find the range of the real number $k$.

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