
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 $C_1: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$) with an eccentricity of $\frac{\sqrt{3}}{3}$, let the left and right vertices be $A_1$ and $A_2$ respectively, and the top and bottom vertices be $B_1$ and $B_2$ respectively. It's given that $\overrightarrow{A_1B_1} \cdot \overrightarrow{A_2B_2} = -1$.  
(1) Find the equation of the ellipse $C_1$.  
(2) Let the left focus of the ellipse $C_1$ be $F_1$, and right focus be $F_2$. A line $l_1$ passes through $F_1$ and is perpendicular to the major axis of the ellipse. A moving line $l_2$ is perpendicular to $l_1$ at point $P$, and the perpendicular bisector of segment $PF_2$ intersects $l_2$ at point $M$. Find the equation of the locus of point $M$, which is $C_2$.  
(3) Suppose $A(x_1, 2)$, $B(x_2, y_2)$, and $C(x_0, y_0)$ are distinct points on the locus $C_2$ from part (2), and $AB \perp BC$. Determine the range of values for $y_0$.

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