
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
"Classification discussion" is an important mathematical thinking method. The following is the process of solving problems using the mathematical thinking of classification discussion. Please read carefully and answer the three questions raised after the problem. Example: Three rational numbers $a$, $b$, $c$ satisfy $abc \gt 0$. Find the value of $\frac{{|a|}}{a}+\frac{{|b|}}{b}+\frac{{|c|}}{c}$.  
Solution: According to the problem, the three rational numbers $a$, $b$, $c$ are all positive numbers or one of them is positive and the other two are negative.  
① When $a$, $b$, $c$ are all positive numbers, i.e., $a \gt 0$, $b \gt 0$, $c \gt 0$,  
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 number and the other two are negative numbers, let $a \gt 0$, $b \lt 0$, $c \lt 0$,  
then: $\frac{{|a|}}{a}+\frac{{|b|}}{b}+\frac{{|c|}}{c}=\frac{a}{a}+\frac{{-b}}{b}+\frac{{-c}}{c}=1+(-1)+(-1)=-1$;  
In summary, the value of $\frac{{|a|}}{a}+\frac{{|b|}}{b}+\frac{{|c|}}{c}$ is $3$ or $-1$.  
Please answer the following questions based on the above problem-solving ideas:  
$(1)$ Given $|a|=3$, $|b|=1$, and $a \lt b$, find the value of $a+b$.  
$(2)$ Given $a$, $b$ are rational numbers, when $ab\neq 0$, find the value of $\frac{a}{{|a|}}+\frac{b}{{|b|}}$.  
$(3)$ Given $a$, $b$, $c$ are rational numbers, $a+b+c=0$, $abc \lt 0$. Find the value of $\frac{{b+c}}{{|a|}}+\frac{{a+c}}{{|b|}}+\frac{{a+b}}{{|c|}}$.

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