
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 a system of two linear equations in $a$ and $b$: $\left\{\begin{array}{l}{a+2b=4}\\{2a+b=3-m}\end{array}\right.$.<br/>$(1)$ Express $a+b$ in terms of $m$.<br/>$(2)$ If the solution of the system satisfies $a-b \gt -4$, find the range of values for $m$.<br/>$(3)$ Under the condition in $(2)$, if $m$ is a positive integer, find the solution to the equation in $x$: $mx-\frac{1-x}{2}=5$.

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