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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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