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Geometry Level 5

If a circle passes through the points of intersection of the coordinate axes with the lines $$\lambda x-y+1=0$$ and $$x-2y+3=0$$, then let the possible values of $$\lambda$$ be $$\lambda_1,\lambda_2,\cdots$$. Calculate $$\left\lceil \displaystyle\prod_{i=1}^n \lambda_i\right\rceil+\left\lfloor \displaystyle\sum_{i=1}^n \lambda_i\right\rfloor$$

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