
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 point \( P \) be a moving point on the circle \( C_1 : x^2 + y^2 = 2 \). Draw a perpendicular line from point \( P \) to the x-axis, and let \( Q \) be the foot of this perpendicular. Point \( M \) satisfies \( \sqrt{2} M Q = P \).

1. Find the equation of the locus \( C_2 \) of point \( M \).
2. From a point \( T \) on the line \( x=2 \), draw two tangents to the circle \( C_1 \), with points of tangency \( A \) and \( B \). If the line \( AB \) intersects the curve \( C_2 \) found in part (1) at points \( C \) and \( D \), find the range of values for \( \frac{|CD|}{|AB|} \).

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