
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
In the plane Cartesian coordinate system \( xOy \), point \( A(0, 3) \), and line \( L: y = 2x - 4 \). Let circle \( C \) have a radius of 1, and the center of circle \( C \) lies on line \( L \). If there exists a point \( M \) on circle \( C \) such that \( |MA| = 2|MD| \), then the range of the x-coordinate \( a \) of the center \( C \) is ___.

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