
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 the curve \( C_{1}: \frac{(x-2)^{2}}{2}+y^{2}= 1 \) and the line \( l: x=-a-\frac{1}{8} \), if there exist four points on the curve \( C_{1} \) such that their distances to the point \(\left(-a+\frac{1}{8}, 0\right)\) are equal to their distances to the line \( l \), find the range of values for \( a \).

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