
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 $M=\left\{(x, y) \mid y=x^{2}\right\}$, $N=\left\{(x, y) \mid x^{2}+(y-a)^{2}=1\right\}$, and $A=M \cap N$:
1. If $|\mathrm{A}|=3$, find the range of the real number $a$.
2. If $A=\emptyset$, find the range of the real number $a$.

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