
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
Let the complex number $\beta=x+yi$ ($x, y \in \mathbb{R}$) correspond to the point $P(x, y)$ on the complex plane.
1. If $\beta$ is a complex root of the quadratic equation in $t$: $t^2-2t+m=0$ ($m \in \mathbb{R}$), and $|\beta|=2$, find the value of the real number $m$;
2. Suppose the complex number $\beta$ satisfies $|\beta+3|+(-1)^n|\beta-3|=3a+(-1)^na$ (where $n \in \mathbb{N}^*$, constant $a \in \left( \frac{3}{2}, 3 \right)$), when $n$ is odd, the trajectory of the moving point $P(x, y)$ is $C_1$. When $n$ is even, the trajectory of the moving point $P(x, y)$ is $C_2$. Both curves pass through point $D(2, \sqrt{2})$, find the equations of trajectories $C_1$ and $C_2$;
3. Under the conditions of (2), there exists a point $A$ on the trajectory $C_2$, such that the minimum distance between point $A$ and point $B(x_0, 0)$ ($x_0 > 0$) is not less than $\frac{2\sqrt{3}}{3}$, find the range of the real number $x_0$.

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