
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 universal set be $U = \mathbf{R}$.
1. Solve the inequality about $x$: $|x-1| + a - 1 \geqslant 0 \quad (a \in \mathbf{R})$.
2. Let $A$ be the solution set of the inequality from part (1). The set $B=\left\{ x \left\lvert\, \sin \left( \pi x - \frac{\pi}{3} \right) + \sqrt{3} \cos \left( \pi x - \frac{\pi}{3} \right) = 0 \right.\right\}$. If $\left(C_{\cup} A\right) \cap B$ has exactly three elements, determine the possible values of $a$.

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