
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 $y=f\left(x\right)$, $x\in A$, for any $a$, $b$, $c\in A$, there exists a triangle with sides $f\left(a\right)$, $f\left(b\right)$, $f\left(c\right)$, then the function is called a triangle function. If the function $f(x)=\left\{\begin{array}{l}{x+m, 0≤x≤\frac{3}{2}}\\{x+\frac{1}{{x-1}}, \frac{3}{2}＜x≤3}\end{array}\right.$ is a triangle function, then the range of real number $m$ is ______.

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