
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
In the rectangular coordinate system $xOy$, two point sets $M$ and $N$ satisfy $M=\left\{(x, y) \mid x^{2}+y^{2}=1\right\}$, $N=\{(x, y) \mid |x-1|+|y-1|=a\}$. If $M \bigcap N=\varnothing$, find the range of the real number $a$.

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