
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 $f\left( x \right)=\frac{b-g\left( x \right)}{a+g\left( x \right)}$ with domain $R$ is an odd function.

(Ⅰ) Find the expression for $f(x)$, determine the monotonicity of $f(x)$ on its domain $R$, and provide a proof;

(Ⅱ) If the equation $f(x)=m$ has a solution in the interval $[-1, 0)$, find the range of $f\left( \frac{1}{m} \right)$.

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