
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
$(1)$ Observation:<br/>Materials: Solve the system of equations $\left\{\begin{array}{l}{x+y=4①}\\{3(x+y)+y=14②}\end{array}\right.$<br/>Substitute the entirety of ① into ②, we get $3\times 4+y=14$,<br/>Solving gives $y=2$,<br/>Substitute $y=2$ into ①, we get $x=2$,<br/>Therefore, $\left\{\begin{array}{l}{x=2}\\{y=2}\end{array}\right.$<br/>This method is called "substitution of the entirety". By observing carefully, many systems of equations can be solved using this method. Please directly write down the solution of the system of equations $\left\{\begin{array}{l}{x-y-1=0①}\\{4(x-y)-y=5②}\end{array}\right.$ as ______.<br/>$(2)$ Practical application: Please use the "substitution of the entirety" method to solve the system of equations $\left\{\begin{array}{l}{2x-3y-2=0①}\\{\frac{2x-3y+5}{7}+2y=9②}\end{array}\right.$;<br/>$(3)$ Extended application: If the solution of the system of linear equations in $x$ and $y$ $\left\{\begin{array}{l}{2x+y=-3m+2}\\{x+2y=7}\end{array}\right.$ satisfies $x+y>-5/6$, find all positive integer values of $m$ that satisfy this condition.

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