
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
Let $a$ be a rational number. Now we use $\{a\}$ to represent the smallest integer not less than $a$, such as $\{4.2\}=5$, $\{-5.3\}=-5$, $\{0\}=0$, $\{-3\}=-3$. Under this rule: any rational number can be written in the form $a=\{a\}-b$, where $0\leqslant b \lt 1$.
$(1)$ Write down directly the relationship between $\{m\}$ and $m$, $m+1$;
$(2)$ Solve the following problems according to the relationship in $(1)$:
① If $\{3x+2\}=8$, find the range of values for $x$;
② Solve the equation: $\{3x-2\}=2x+\frac{1}{2}$.

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