
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
Consider the region $Ω_n$ (including the boundary) enclosed by the ellipse $\frac{x^{2}}{4}+ \frac{ny^{2}}{4n+1}=1$ where $n=1,2,3,...$. When the point $(x,y)$ is on $Ω_1$, $Ω_2$, $...$ respectively, the maximum value of $x+y$ is $M_1$, $M_2$, $...$ respectively. Find $\lim_{n \to +\infty }M_{n}$ = \_\_\_\_\_\_.

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