
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 $f(x)=|ax+1|$ $(a\in\mathbb{R})$, the solution set of the inequality $f(x)\leqslant 3$ is $\{x|-2\leqslant x\leqslant 1\}$.  
(I) Find the value of $a$;  
(II) If $|f(x)-2f\left( \frac{x}{2}\right)|\leqslant k$ always holds, find the range of $k$.

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