The vacuum system contains a wall with possibly leaking components. The physicists have performed vacuum leak tests on some of these components by flushing them with helium gas and then noting down whether their mass spectrometer detected any spike in helium in the vacuum system directly following this release of gas. If the component has even the tiniest leak they will detect it this way but there are some complications as well. The helium will rise up and spread out from where they released it and if it passes by any other leaking component, that will also trigger a positive reading. For each unit distance the helium has risen it will also have expanded by one unit. Thus the leak test will produce a positive result if the tested component is leaking or if there is a leaking component above it for which the $x$ coordinates differs by at most half of the difference in the $y$ coordinate. See Figure 1 for an example.
You start out with a positive mindset thinking that there are probably just a few leaking components responsible for all the positive measurements. To determine if this is indeed possible you set out to determine the minimum number of leaking components that could give rise to the observed leak test results.
The first line of input contains an integer $n$ ($1 \le n \le 2 \cdot 10^5$), the number of components involved. The following $n$ lines each contain two integers $x$ and $y$ and a character $c$ ($-10^8 \le x,y \le 10^8$, $c \in \{ \texttt{-, P, N}\} $), where $(x, y)$ are the coordinates of a component and $c$ describes a possible leak test result, with the following meanings:
‘-’ – No leak test has been performed on this component
‘N’ – Leak test gave negative response on this component
‘P’ – Leak test gave positive response on this component
No two components have the same position.
Output a single integer, the minimum number of leaking components that could give rise to the observed leak test results. If no set of leaking components could give rise to the observed results, instead output the single word impossible.
Sample Input 1 | Sample Output 1 |
---|---|
4 1 -1 P 2 2 P -1 3 N -2 -1 - |
1 |
Sample Input 2 | Sample Output 2 |
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2 0 0 N 1 2 P |
impossible |