
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
Given that $a$, $b$, and $c$ are non-zero rational numbers, please investigate the following questions:<br/>
$(1)$ When $a \lt 0$, $\frac{a}{|a|}=$______;<br/>
$(2)$ The minimum value of $\frac{ab}{|ab|}+\frac{|bc|}{bc}+\frac{ca}{|ca|}+\frac{|abc|}{abc}$ is______.

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