
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
Reverse the opening of the quadratic function $y=-2x^2+8x-5$ and shift it up or down to obtain a new parabola. The new parabola has one intersection point with the line $y=kx+1$ at $(3,4)$. Find:  
(1) The equation of this new parabola;  
(2) The other intersection point between this new parabola and the line $y=kx+1$.

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