
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 \(a \in \mathbf{R}^{+}\), the sets \(M = \{(x, y) \mid x^{2} + y^{2} = a^{2}\}\) and \(N = \{(x, y) \mid |x + y| + |x - y| \leq 2\}\), determine the necessary and sufficient condition for \(M \subseteq N\).

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