
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 $f(x)=x^{3}-3x+2+m$ ($m > 0$), there exist three distinct real numbers $a$, $b$, $c$ in the interval $[0,2]$ such that the triangle with side lengths $f(a)$, $f(b)$, and $f(c)$ is a right-angled triangle. The range of $m$ is \_\_\_\_\_\_.

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