
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 functions $f(x)=-x^{2}+6x+m$ and $g(x)=2\sin(2x+\frac{π}{3})$. For any $x_{0}∈[0,\frac{π}{4}]$, there exist $x_{1}$ and $x_{2}\in \left[-1,3\right]$, such that $f(x_{1})\leqslant g(x_{0})\leqslant f(x_{2})$. The range of real number $m$ is ______.

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