
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: Algebraic expression $x^{2}-4x+3$.  
$(1)$ When $x=2$, the value of the algebraic expression $x^{2}-4x+3$ is ______;  
$(2)$ When the value of the algebraic expression $x^{2}-4x+3$ is $4$, the values of $x$ are $x_{1}$ and $x_{2}$. Find the value of ${x}_{1}^{2}+{x}_{2}^{2}$;  
$(3)$ Given that when $x=m+2$, the value of the algebraic expression $x^{2}-4x+3$ is $n_{1}$; when $x=2m+1$, the value of the algebraic expression $x^{2}-4x+3$ is $n_{2}$. If the value of $\frac{n_2}{n_1}$ is an integer, find the value of the positive integer $m$.

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