
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 for any non-zero real numbers \(a\) and \(b\), the inequality \(|3a+b|+|a-b|\geq |a|(|x-1|+|x+1|)\) always holds, find the range of the real number \(x\).  
**(2)** Given that the solution set of the inequality \(|2x-1| < 1\) is \(M\), and if \(a, b \in M\), compare the sizes of \( \frac{1}{ab}+1\) and \( \frac{1}{a}+\frac{1}{b}\), and explain the reason.

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