
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 $f\left(x\right)=\log _{2}(x^{2}-4ax+3)$.  
$(Ⅰ)$ When $a=1$, find the domain and the interval where $f\left(x\right)$ is decreasing.  
$(Ⅱ)$ If the function $f\left(x\right)$ is increasing on $\left(1,+\infty \right)$, find the range of real numbers for $a$.

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