
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 the function $f(x)=|ax-1|$,  
(1) If the solution set for $f(x) \leqslant 2$ is $[-3,1]$, find the value of the real number $a$;  
(2) If $a=1$ and there exists $x\in \mathbb{R}$ such that the inequality $f(2x+1)-f(x-1) \leqslant 3-2m$ holds, find the range of values for the real number $m$.

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