
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
$(1)$ If the solution of the system of linear equations in two variables $x$ and $y$, $\left\{\begin{array}{l}2x+y=-3m+2\\ x+2y=4\end{array}\right.$, satisfies $x+y>-3/2$, find all positive integer values of $m$ that satisfy the condition.<br/>$(2)$ Given that $a$, $b$, and $c$ are the three side lengths of $\triangle ABC$, satisfying $a^{2}+b^{2}=10a+8b-41$, and $c$ is the longest side of $\triangle ABC$. Find the range of values for $c$.

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