
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 an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ and two lines $l_1: y = \frac{1}{2}x$ and $l_2: y = -\frac{1}{2}x$, parallel lines to $l_1$ and $l_2$ pass through a point $P$ on the ellipse and meet $l_1$ and $l_2$ at points $M$ and $N$, respectively. If $|MN|$ is a constant value, then the value of $\sqrt{\frac{a}{b}}$ is ______.

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