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The traffic on the Internet is increasing these days due to smartphones. The wireless carriers have to enhance their network infrastructure.
The network of a wireless carrier consists of a number of base stations and lines. Each line connects two base stations bi-directionally. The bandwidth of a line increases every year and is given by a polynomial f(x) of the year x.
Your task is, given the network structure, to write a program to calculate the maximal bandwidth between the 1-st and N-th base stations as a polynomial of x.
The input consists of multiple datasets. Each dataset has the following format:
N M u1 v1 p1 . . . uM vM pM
The first line of each dataset contains two integers N (2 ≤ N ≤ 50) and M (0 ≤ M ≤ 500), which indicates the number of base stations and lines respectively. The following M lines describe the network structure. The i-th of them corresponds to the i-th network line and contains two integers ui and vi and a polynomial pi . ui and vi indicate the indices of base stations (1 ≤ ui , vi ≤ N); pi indicates the network bandwidth.
Each polynomial has the form of:
aLxL + aL−1xL−1 + ... + a2x2 + a1x + a0
where L (0 ≤ L ≤ 50) is the degree and ai ’s (0 ≤ i ≤ L, 0 ≤ ai ≤ 100) are the coefficients. In the input,
For example, 2x2 + 3x + 5 is represented as 2x^2+3x+5; 2x3 + x is represented as 2x^3+x, not 2x^3+0x^2+1x+0 or the like. No polynomial is a constant zero, i.e. the one with all the coefficients being zero.
The end of input is indicated by a line with two zeros. This line is not part of any dataset.
For each dataset, print the maximal bandwidth as a polynomial of x. The polynomial should be represented in the same way as the input format except that a constant zero is possible and should be represented by “0” (without quotes).
3 3 1 2 x+2 2 3 2x+1 3 1 x+1 2 0 3 2 1 2 x 2 3 2 4 3 1 2 x^3+2x^2+3x+4 2 3 x^2+2x+3 3 4 x+2 0 0
2x+3 0 2 x+2