Description

We simplify this model. Initially, there are $I$ infectives and $S$ susceptibles. For each day, suppose there are currently $I_i$ infectives, $S_i$ susceptibles, and $R_i$ recovered. Then each day, $\lceil \beta S_iI_i \rceil$ people will be infected (changing from susceptible to infective), and $\lceil \gamma I_i \rceil$ people will be cured (changing from infective to recovered).

Here, $\beta$ is the infection rate, $\gamma$ is the recovery rate, and $\lceil \ \rceil$ is the ceiling function.

Find how many susceptibles $S$, infectives $I$, and recovered $R$ there are after $n$ days.

Note: Infectives and recovered are settled daily, and the result depends only on the values at the start of that day. That is, someone who recovers on that day does not affect how many people they infect on the same day.

If the computed number of newly infected people exceeds the number of susceptibles, then all susceptibles become infected.

Input Format

The first line contains three positive integers, representing the number of susceptibles on day $0$, $S_0$, the number of infectives on day $0$, $I_0$, and the number of days $n$ (initially the number of recovered is $R_0=0$).

The second line contains two floating-point numbers, representing the infection rate $\beta$ and the recovery rate $\gamma$.

Output Format

Output one line with three integers, representing the numbers of susceptibles $S$, infectives $I$, and recovered $R$ after $n$ days.

980 20 2
0.0005 0.00001

955 43 2

1400000000 1 10
0.000000003 0.001

1386791252 13205592 3157

1919 810 1
0.00001 0.1

1903 745 81

Hint

For $30\%$ of the testdata, $n=1$. For another $30\%$ of the testdata, $S_0, R_0 \le 10^4$. For $100\%$ of the testdata, $1 \le S_0+R_0 \le 2\times 10^9, 0 < \beta, \gamma < 1, 1 \le n \le 100$.

Translated by ChatGPT 5