
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
For real numbers $a$ and $b$, define the operation "$*$" as follows: $$a*b= \begin{cases} a^{2}-ab & \text{if } a\leq b \\ b^{2}-ab & \text{if } a>b \end{cases}$$ Let $f(x) = (2x-1)*(x-1)$, and suppose the equation $f(x)=m$ ($m \in \mathbb{R}$) has exactly three distinct real roots $x_1$, $x_2$, $x_3$. Then, the range of values for the real number $m$ is \_\_\_\_\_\_; the range of values for $x_1+x_2+x_3$ is \_\_\_\_\_\_.

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