
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
Define $S = \{(x,y) : (x - M)^2 + \frac{(y-N)^2}{2} \leq M^2\}$, where $M = t - [t]$ and $N = [t] + 0.5$. Compute the area of $S$, and determine if the point $(0,0)$ belongs to $S$ for all $t \geq 0$.
- **A)** $\pi \sqrt{2}$
- **B)** $\pi M^2 \sqrt{2}$ 
- **C)** $\pi M^2$
- **D)** $2\pi M$

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