
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 that the graph of the exponential function $y=g(x)$ passes through the point $(2,4)$, and the function $f(x) = \frac{b-g(x)}{a+g(x)}$ defined on the domain $\mathbb{R}$ is an odd function.  
(1) Find the expression for $f(x)$, determine the monotonicity of $f(x)$ on the domain $\mathbb{R}$, and provide a proof;  
(2) If the equation $f(x)=m$ has a solution in the interval $[-1,0)$, find the range of values for $f\left(\frac{1}{m}\right)$.

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