
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 $C$: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$, its left and right foci are $F_{1}$ and $F_{2}$, respectively. Point $M$ is on the ellipse, and $|MF_{1}| + |MF_{2}| = 4$. The eccentricity of the ellipse is $e = \frac{1}{2}$;  
(1) Find the standard equation of ellipse $C$;  
(2) Given $N(4,0)$, draw a line $l$ with slope $K (K > 0)$ through point $N$ that intersects the ellipse at two distinct points $A$ and $B$. The perpendicular bisector of segment $AB$ is $l'$, and the y-intercept of $l'$ is $m$. Find the range of $m$.

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