
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
Reading comprehension:<br/>Definition: The values of unknowns that satisfy both equations and inequalities are called "ideal solutions" of these equations and inequalities. For example, given the equation $2x-1=1$ and the inequality $x+1 \gt 0$, when $x=1$, $2x-1=2\times 1-1=1$ and $1+1=2 \gt 0$ both hold true. In this case, "$x=1$" is considered an "ideal solution" of the equation $2x-1=1$ and the inequality $x+1 \gt 0".<br/>Problem solving:<br/>$(1)$ Please determine which of the following inequalities (or sets of inequalities) the solution of the equation $4x-3=1$ is an "ideal solution" to ______ (write the corresponding numbers directly);<br/>①$x-3 \gt 3x-1$<br/>②$4\left(x-1\right)\leqslant 2$<br/>③$\left\{\begin{array}{c}x+2＞0\\ 3x-3≤1\end{array}\right.$<br/>$(2)$ If $\left\{\begin{array}{l}x=m\\ y=n\end{array}\right.$ is an "ideal solution" of the system of equations $\left\{\begin{array}{l}x+2y=6\\ 2x+y=3q\end{array}\right.$ and the inequality $x+y \gt 1$, find the range of values for $q$;<br/>$(3)$ When $k \lt 3$, the solutions of the equation $3\left(x-1\right)=k$ are all "ideal solutions" of this equation and the inequality $4x+n \lt x+2m". If $m+n\geqslant 0$ and there is only one integer $n$ that satisfies the condition, find the range of values for $m$.

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