
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
Problem: For positive numbers $a$ and $b$ satisfying $a+b=1$, find the minimum value of $\frac{1}{a}+\frac{2}{b}$. One solution is: $\frac{1}{a}+\frac{2}{b}=({\frac{1}{a}+\frac{2}{b}})({a+b})=1+\frac{b}{a}+\frac{{2a}}{b}+2≥3+2\sqrt{2}$, and the equality holds when $\frac{b}{a}=\frac{{2a}}{b}$ and $a+b=1$, i.e., $a=\sqrt{2}-1$ and $b=2-\sqrt{2}$. Learn from the above solution and solve the following problems:<br/>$(1)$ If positive real numbers $x$ and $y$ satisfy $xy=3x+y$, find the minimum value of $x+y$;<br/>$(2)$ If positive real numbers $a$, $b$, $x$, $y$ satisfy $\frac{{{x^2}}}{{{a^2}}}-\frac{{{y^2}}}{{{b^2}}}=1$ and $a \gt b$, compare $a^{2}-b^{2}$ and $\left(x-y\right)^{2}$, and explain the reason;<br/>$(3)$ If $m \gt 0$, using the conclusion from $(2)$, find the minimum value of the algebraic expression $M=\sqrt{3m-5}-\sqrt{m-2}$, and determine the value of $m$ that minimizes $M$.

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