
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 $f\left(x\right)=\lg (x^{2}-ax+4)$ and the power function $g\left(x\right)=x^{a}$. $p$: the domain of $f\left(x\right)$ is $R$, $q$: $g\left(x\right)$ is monotonically increasing on $\left(0,+\infty \right)$. <br/>$(1)$ If $p\wedge q$ is true, find the range of real number $a$; <br/>$(2)$ If $p\vee q$ is true and $p\wedge q$ is false, find the range of real number $a$.

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