
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 $C$ with the equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ $(a > b > 0)$, its eccentricity is $\frac{\sqrt{2}}{2}$, vertices $A(a, 0)$, $B(0, b)$, and the distance from the center $O$ to the line $AB$ is $\frac{2}{\sqrt{3}}$.  
(1) Find the equation of ellipse $C$.  
(2) Let point $P$ on ellipse $C$ satisfy $\vec{OP} = \lambda \vec{OM} + 2\mu \vec{ON}$, where $M$, $N$ are points on ellipse $C$, and the product of the slopes of lines $OM$ and $ON$ is $-\frac{1}{2}$. If $Q(\lambda, \mu)$ is a movable point, and $E_1(-\frac{\sqrt{3}}{2}, 0)$, $E_2(\frac{\sqrt{3}}{2}, 0)$ are two fixed points, find the value of $|QE_1| + |QE_2|$.

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