
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
Classifying cases is an important mathematical method. For example, when simplifying $|a|$, we can classify it as follows: when $a \gt 0$, $|a|=a$; when $a=0$, $|a|=0$; when $a \lt 0$, $|a|=-a$. Solve the following problems using this method:<br/>$(1)$ When $a=5$, find the value of $\frac{|a|}{a}$.<br/>$(2)$ When $a=-2$, find the value of $\frac{a}{|a|}$.<br/>$(3)$ Given that $a$ and $b$ are rational numbers, when $ab \gt 0$, try to find the value of $\frac{a}{|a|}+\frac{|b|}{b}$.<br/>$(4)$ Given that $a$ and $b$ are rational numbers, when $abc \lt 0$, try to find the value of $\frac{a}{|a|}+\frac{b}{|b|}+\frac{c}{|c|}+\frac{abc}{|abc|$.

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