
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 a parabola $y = -x^2 + bx + c$ intersects the x-axis at two points A$(x_1, 0)$ and B$(x_2, 0)$ (A is to the left of B), and $x_1 + x_2 = 4$.

(1) Find the value of $b$ and the range of $c$;  
(2) If $AB = 2$, find the equation of the parabola;  
(3) Let the intersection of this parabola with the y-axis be C, the vertex be D, and the intersection of the axis of symmetry with the x-axis be E. Is there such a parabola that makes $\triangle AOC \cong \triangle BED$? If it exists, find the equation of the parabola; if not, explain why.

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