
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 $f(x)=|x-a|$, where $a\in R$.

(1) If $a=1$, solve the inequality $f(x)\geqslant \frac{1}{2}(x+1)$;

(2) Let the range of the function $g(x)=f(x)-|x-2|$ be $A$. If $A\subseteq [-1,3]$, find the range of values for $a$.

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