
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
In the process of solving mathematical problems, we often use the mathematical concept of "classification and discussion." The following is the process of solving problems using this concept. Please read carefully and answer the questions raised at the end.<br/>【Question】Three rational numbers $a$, $b$, $c$ satisfy $abc \gt 0$. Find the value of $\frac{|a|}{a}+\frac{|b|}{b}+\frac{|c|}{c}$. <br/>【Solution】<br/>Solution: According to the given conditions, the three rational numbers $a$, $b$, $c$ are all positive or one is positive and the other two are negative.<br/>① When $a$, $b$, $c$ are all positive, i.e., $a \gt 0$, $b \gt 0$, $c \gt 0$, <br/>then: $\frac{|a|}{a}+\frac{|b|}{b}+\frac{|c|}{c}=\frac{a}{a}+\frac{b}{b}+\frac{c}{c}=1+1+1=3$; ② When $a$, $b$, $c$ have one positive and the other two negative, let $a \gt 0$, $b \lt 0$, $c \lt 0$, <br/>then: $\frac{|a|}{a}+\frac{|b|}{b}+\frac{|c|}{c}=\frac{a}{a}+\frac{-b}{b}+\frac{-c}{c}=1-1-1=-1$<br/>Therefore, the value of $\frac{|a|}{a}+\frac{|b|}{b}+\frac{|c|}{c}$ is $3$ or $-1$.<br/>【Exploration】Based on the above solution, answer the following questions:<br/>$(1)$ If three rational numbers $a$, $b$, $c$ satisfy $abc \lt 0$, find the value of $\frac{|a|}{a}+\frac{|b|}{b}+\frac{|c|}{c}$;<br/>$(2)$ Given $|a|=9$, $|b|=4$, and $a \lt b$, find the value of $a-2b$.

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