Mixture

Serge is the chef of the famous restaurant "Salt, Pepper & Garlic". He wants to earn his first Michelin star. He has been told that a secret expert will visit the restaurant this evening.

The expert's name is not disclosed, but Serge knows which dish the expert will order and what the expert likes. The expert requires a very precise ratio of salt, pepper, and garlic powder in the dish.

Serge keeps bottles of mixtures of salt, pepper, and garlic powder on a special shelf. For each bottle, he knows the exact amount of each ingredient in kilograms. He can combine any number of bottled mixtures, or use one bottle directly, to get a mixture with the ratio the dish needs.

The amount of mixture added to a dish is very small, so you may assume the bottles always contain enough ingredients. The ratio values can be very large, however.

Serge wants to know whether he can make the expert's favourite mixture from the bottles on the shelf and, if so, the minimum number of bottles needed. The shelf changes over time as Serge receives new bottles or lends his bottles to other chefs, so he needs the answer after each change.

Table 1: Bottles on the shelf

The expert's favourite mixture is 1 : 1 : 1, and the shelf has three bottles (Table 1). Bottles 1 and 2 used in equivalent amounts are enough to make the desired mixture. If bottle 2 is removed, the desired mixture can no longer be made.

The first line contains three non-negative integers $S_f$, $P_f$, and $G_f$ ($0 \le S_f, P_f, G_f$, $0 < S_f + P_f + G_f \le 10^6$), the amounts of salt, pepper, and garlic powder in the expert's favourite mixture. For any real $\alpha > 0$, $(\alpha S_f, \alpha P_f, \alpha G_f)$ is also a favourite mixture.

The second line contains a positive integer $N$ ($N \le 100000$), the number of changes to the shelf. The shelf is empty at the start. Each of the next $N$ lines describes one change.

• To add a bottle, the line starts with the letter A, followed by three non-negative integers $S_i$, $P_i$, and $G_i$ ($0 \le S_i, P_i, G_i$, $0 < S_i + P_i + G_i \le 10^6$). Bottles are numbered consecutively from 1, so the $i$-th added bottle has number $i$.
• To remove a bottle, the line starts with the letter R, followed by the bottle number $r_i$. All removed numbers are distinct, and $r_i$ never exceeds the number of bottles added so far.

Print $N$ lines. The $j$-th line contains the minimum number of bottles needed to make a mixture with the expert's favourite proportions, using the bottles on the shelf after the first $j$ changes. Print 0 if the mixture cannot be made.

Bottles 1 and 3 contain salt, pepper, and garlic powder in the same proportions.