
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
$(1)$ Given $x \gt -1$, find the minimum value of the function $y=\frac{{({x+2})({x+3})}}{{x+1}}$, and determine the value of $x$ when the minimum value is achieved;<br/>$(2)$ Problem: For positive numbers $a$ and $b$ such that $a+b=1$, find the minimum value of $\frac{1}{a}+\frac{2}{b}$. One possible 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 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 problem: If real numbers $a$, $b$, $x$, $y$ satisfy $\frac{{{x^2}}}{{{a^2}}}-\frac{{{y^2}}}{{{b^2}}}=1$, compare $a^{2}-b^{2}$ and $\left(x-y\right)^{2}$, and indicate the conditions under which equality holds;<br/>$(3)$ Using the conclusion from $(2)$, find the minimum value of $M=\sqrt{4m-3}-\sqrt{m-1}$, and determine the value of $m$ that minimizes $M$.

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