
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 $f(x)=\frac{x}{x-a}$.<br/>$(1)$ If $a \gt 0$ and $f\left(x\right)$ is monotonically decreasing in $\left(1,+\infty \right)$, find the range of values for $a$;<br/>$(2)$ The necessary and sufficient condition for the graph of the function $y=g\left(x\right)$ to be symmetric with respect to the point $P\left(m,n\right)$ is that the function $y=g\left(x+m\right)-n$ is an odd function. When $a=1$, find the center of symmetry of $h\left(x\right)=f\left(x\right)+x^{3}-3x^{2}$.

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