
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
Let the function $f(x) = |x-1| + |x-a| (a \in \mathbb{R})$.

$(1)$ If $a = -3$, find the minimum value of the function $f(x)$;

$(2)$ If for all $x \in \mathbb{R}$, $f(x) \leqslant 2a + 2|x-1|$, find the range of values for $a$.

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