
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 sets
$$
\begin{array}{l}
A=\left\{(x, y) \mid (x-2)^{2}+(y-1)^{2} \leqslant 1\right\}, \\
B=\{(x, y) \mid 2|x-1|+|y-1| \leqslant a\}, \\
A \subseteq B.
\end{array}
$$

Determine the range of the real number $a$.

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