
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
$(1)$ Given that $a$, $b$, and $c$ are rational numbers, $a^{2}=9$, $|b|=4$, $c^{3}=27$, and $ab \lt 0$, $bc \gt 0$, find the value of $ab-bc+ca$. <br/>$(2)$ When the range of $x$ is ______, the expression $|x+1|+|x-3|+|x-7|+|x-11|$ has a minimum value of ______. <br/>$(3)$ When the range of $x$ is ______, the expression $|x+1|-|x-7|$ has a maximum value of ______.

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