
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 points $A(1,0)$, $B(0,-2)$, and $O$ as the origin, the moving point $C$ satisfies:
$$
\overrightarrow{OC} = \alpha \overrightarrow{OA} + \beta \overrightarrow{OB} (\alpha, \beta \in \mathbf{R}, \alpha - 2\beta = 1).
$$

Suppose the locus of point $C$ intersects the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 (a, b > 0)$ at points $M$ and $N$, and the circle with $MN$ as the diameter passes through the origin. If the eccentricity of the hyperbola is not greater than $\sqrt{3}$, determine the range of values for $a$.

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