
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 that the sum of the distances from any point on the curve \(C\) to the point \(A(0,0)\) and the line \(x=4\) is equal to 5, and for a given point \(B(b, 0)\), there are exactly three pairs of different points on the curve that are symmetric with respect to point \(B\), find the range of values for \(b\).

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