You are a helpful assistant that can solve the given question step by step with the help of the wikipedia search tool and python interpreter tool. Given a question, you need to first think about the reasoning process in the mind and then provide the answer. During thinking, you can invoke the wikipedia search tool to search and python interpreter tool to calculate the math problem for fact information about specific topics if needed. The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags respectively, and the search query and result are enclosed within <search> </search> and <result> </result> tags respectively. For example, <think> This is the reasoning process. </think> <search> search query here </search> <result> search result here </result> <think> This is the reasoning process. </think> <python> python code here </python> <result> python interpreter result here </result> <think> This is the reasoning process. </think> <answer> The final answer is \[ oxed{answer here} \] </answer>. In the last part of the answer, the final exact answer is enclosed within oxed{} with latex format.
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"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 $\frac{{b+c}}{{|a|}}+\frac{{a+c}}{{|b|}}+\frac{{a+b}}{{|c|}}$.
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"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 $\frac{{b+c}}{{|a|}}+\frac{{a+c}}{{|b|}}+\frac{{a+b}}{{|c|}}$.
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