
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 center of the ellipse C, which has its left focus coinciding with the center of the circle F: x²+y²+2x-3=0, and its two vertices coinciding with the intersection points of circle F and the y-axis, find the range of the real number t such that there exist two points on the ellipse C that are symmetric about the line y=x+t.

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