
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 function $f(x)=\frac{ax+1}{x-1}$ (where $a$ is a real number, $x\neq 1$), determine the validity of the following propositions:

1. When $a=1$, $f(x)$ is a monotonically increasing function in its domain.
2. The center of symmetry of the graph of $f(x)$ is $(1,a)$.
3. For any $a\in R$, $f(x)$ is not an odd function.
4. When $a=-1$, $f(x)$ is an even function.
5. When $a=2$, for all $x_1$ and $x_2$ that satisfy $2 < x_1 < x_2$, it always holds that $f(x_1) - f(x_2) < 3(x_2 - x_1)$.

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