
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 two sets of points in a plane: 

\[ M = \{(x, y) \mid x + y + 1 \geq \sqrt{2(x^2 + y^2)}, x, y \in \mathbb{R} \} \]
\[ N = \{(x, y) \mid |x - a| + |y - 1| \leq 1, x, y \in \mathbb{R} \} \]

If \( M \cap N \neq \emptyset \), what is the range of the value of \( a \)?

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