
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 $f(x)=\log_2(x^2-ax+3a)$.<br/>$(1)$ If the range of $f(x)$ is $R$, find the range of real number $a$;<br/>$(2)$ Given $g(x)=m\cdot 3^{x}+5-2m$, when $a=4$, if for any $x_1∈[1,2+2\sqrt{6}]$, there exists $x_{2}\in [0,2]$ such that $f(x_{1})=g(x_{2})$ holds, find the range of real number $m$.

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