
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 expressions $A=\frac{x}{y+z}$, $B=\frac{y}{x+z}$, $C=\frac{z}{x+y}$.  
$(1)$ If $x$, $y$, $z$ are positive integers and $x \gt y \gt z$, then the relationship between $A$, $B$, $C$ is ______;  
$(2)$ If $x=y=1$, and $z$ is a real root of the equation $m^{2}-2023m+1=0$, then $\frac{1}{A}+\frac{1}{B}+\frac{1}{C}=\_\_\_\_\_\_$.

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