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# Problem HDigital Speedometer

A digital speedometer shows a vehicle’s speed as integer miles per hour. There are occasions when the sensed speed varies between two integer values, such as during cruise control. Using a single threshold to round between adjacent integers often makes the display toggle rapidly between the two integers, which is distracting to the driver.

Your team must implement a smoothing technique for the display using separate rising and falling thresholds ($t_ r$ and $t_ f$, $t_ f < t_ r$, respectively). See Figure 1 for a graphical depiction of the Sample Input for use with the following rules. Each sensed speed, $s$, falls between two adjacent integers $i$ and $j$, $i \le s < j$, where $j = i + 1$. When displaying the sensed speed $s$ as an integer:

• When $s$ falls between $i$ and $i+t_ f$, $s$ is displayed as $i$.

• When $s$ falls between $i+t_ r$ and $j$, $s$ is displayed as $j$.

• When $s$ falls between $i+t_ f$ and $i+t_ r$, $s$ is displayed as $i$ if the most recent preceding value for $s$ outside of range $[i+t_ f, i+t_ r]$ is less than $i+t_ r$, and $s$ is displayed as $j$ if the most recent preceding value for $s$ outside of range $[i+t_ f, i+t_ r]$ is greater than $i+t_ r$.

• Any sensed speed, $0 < s < 1$, must display as $1$ because any non-zero speed, no matter how small, must display as non-zero to indicate that the vehicle is in motion.

## Input

The first line of input contains $t_ f$, the falling threshold. The second line of input contains $t_ r$, the rising threshold. The speed sensor reports $s$ in increments of $0.1$ mph. The thresholds are always set halfway between speed increments. All remaining lines until end-of-file are successive decimal speeds, $s$, in miles per hour, one speed per line. The third line of input, which is the first measured speed, will always be $0$. There are at most $1000$ observed speeds $s$ in input.

$0 < t_ f,t_ r < 1; \ \ \ \ t_ f < t_ r; \ \ \ \ 0 \le s \le 120$

## Output

Output is the list of speeds, one speed per line, smoothed to integer values appropriate to $t_ f$ and $t_ r$.

## Sample Explanation

 Input Output Explanation 0.25 Value of $t_ f$. 0.75 Value of $t_ r$. 0 0 Initial input. 2.0 2 Input greater than $0$, below threshold of $2.25$. 5.7 5 Input greater than $2.0$, in threshold range. 5.8 6 Input greater than $2.0$, exceeds upper threshold of $5.75$. 5.7 6 Input less than $5.8$, in threshold range. 5.2 5 Input less than $5.8$, below threshold of $5.25$. 5.7 5 Input greater than $5.2$, in threshold range. 0.8 1 Input greater than $0$ and less than $1$. 0.2 1 Input greater than $0$ and less than $1$.
Sample Input 1 Sample Output 1
0.25
0.75
0
2.0
5.7
5.8
5.7
5.2
5.7
0.8
0.2

0
2
5
6
6
5
5
1
1