GUESSN2

Judge has chosen an integer x in the range [1, n]. Your task is to find x, using query as described below. But be careful, because the Judge is a liar. Judge is allowed to lie up to w times in single game and only in reply for query.

Query

Single query should contains set S = {a₁, a₂, ..., a_k}. The reply for query is "YES", if x is in S. Otherwise the reply is "NO".

1 ≤ k < n

1 ≤ a₁ < a₂ < ... < a_k ≤ n

Communication

You should communicate with Judge using standard input and output.

Attention: the program should clear the output buffer after printing each line. It can be done using fflush(stdout) command or by setting the proper type of buffering at the beginning of the execution - setlinebuf(stdout).

The first line of input contains single integer T, the number of games. Then T games follow.

At the beggining of each game You should send to the Judge a line with command "START_GAME". The Judge will answer You with numbers n, w, m, where n, w are as described above and m is the maximum number of queries that You can use in this game.

Then You should send some queries, every query is a line with "QUERY" keyword, then single-space separated values k a₁ a₂ ... a_k. After each query the Judge will answer "YES" or "NO".

At the end of game You should give answer: "ANSWER y", where 0 ≤ y ≤ n; y=0 means, that You skip this game without the correct answer. Otherwise y is the answer for the game. When y ≠ x, the solution will got WA.

Then start the next game (if there is any).

Scoring

Total score is the sum of scores of single games. If You use c queries in game and You find the x value, Your score is c². If You skip the game, the score is 4m². The smaller score is the better score.

Constraints

1 ≤ T ≤ 2⁷

2 ≤ n ≤ 2¹⁷

2 ≤ w ≤ 2⁴

1 ≤ w * n ≤ 2¹⁹

Example

The example of communication. J=Judge, P=Player.

J: 3

P: START_GAME
J: 2 2 10

P: QUERY 1 1
J: YES
P: QUERY 1 1
J: YES
P: QUERY 1 1
J: YES
P: ANSWER 1

P: START_GAME
J: 2 4 10

P: QUERY 1 2
J: YES
P: QUERY 1 2
J: YES
P: QUERY 1 1
J: YES
P: QUERY 1 1
J: YES
P: QUERY 1 2
J: YES
P: QUERY 1 2
J: YES
P: QUERY 1 2
J: NO
P: QUERY 1 1
J: NO
P: ANSWER 2

P: START_GAME
J: 12345 7 100

P: ANSWER 0

Explanation:

In 1st game Judge said 3 times, that his number is 1 and he didn't lie. The answer is 1, because he can lie only 2 times.

In 2nd game the Judge lied in 3rd, 4th and 7th query.

In 3rd game the Player gave up. The score is 4 * 100².

The score is 3² + 8² + 4*100² = 9 + 64 + 40000= 40073

Note

Be careful. The Judge will try to maximize the number of queries that You will ask. If necessary, the Jugde can also replace chosen value x with the other one. But don't worry too much - at the end of the game, the value x chosen by Judge will satisfy all except at most w of Your queries.

Note 2

If You got SIGXFSZ error, You probably use unnecessary numbers in queries. Let's see at the example:

P: START_GAME
J: 16 2 14

P: QUERY 2 1 2
J: NO
P: QUERY 3 1 2 3
J: NO
P: QUERY 4 1 2 3 4
J: NO
P: QUERY 5 1 2 3 4 5
J: NO
P: QUERY 6 1 2 3 4 5 6
J: NO

In 4th query, there are unnecessary numbers 1 and 2. This numbers cannot be the answer for this game, because the Judge said three times (in 1st, 2nd and 3rd query's reply) "1 and 2 are not OK!", but the Jugde can lie only 2 times. From the same reason in 5th query, the unnecessary numbers are 1, 2 and 3. When n is big enough, the profit from this optimization is huge (and probably SIGXFSZ won't appear).

Similar problems

There is a family of similar problems. Here is the table with them:

subset - the query is about any subset of {1,2,...,n}
range - the query is about any range [a,b]
 interactive - the Judge replies after every query
non-interactive - all queries have to be asked at once, before any reply