Coin Catching Challenge

In a 2D coordinate system, there are \(n\) coins labeled from \(0\) to \(n-1\). Initially, the \(i\)-th coin is at the position \((x_i, y_i)\). Every second, each coin moves 1 unit downwards, i.e. a coin located at \((x,y)\) at some time will be at \((x, y-t)\) after \(t\) seconds. Initially, FJ is at \((0,0)\) and he can move horizontally by at most 1 unit per second (or choose not to move). If at any moment a coin’s position coincides with FJ’s position, FJ catches that coin. Determine whether FJ can catch all coins. If he can, output Abletocatch; otherwise, output Notabletocatch.

Notes:

• The catch time for a coin initially at \((x_i, y_i)\) is exactly \(y_i\), since the coin falls 1 unit per second.
• FJ must be at \(x_i\) at time \(y_i\) to catch the coin.
• Since coins fall at a constant rate, the order in which coins must be caught is fixed by increasing \(y_i\) (i.e. the coin with the smaller \(y_i\) comes first).
• If two coins have the same \(y_i\) but different \(x_i\), it is impossible to catch both.

The problem can be solved by simulating FJ's horizontal movement. Start at \(t=0\) and position \(0\). Sort the coins by \(y_i\). For each coin, check if FJ can move from his last position to the coin's \(x_i\) in the available time (which is the difference in the coin's catch time and the current time). If \(|x_i - \text{current_position}|\) exceeds the available time, output Notabletocatch; otherwise, update FJ's position and time to that coin's position and catch time. If FJ can catch all coins, output Abletocatch.

inputFormat

The first line contains an integer \(n\) \( (1 \le n \le 10^5)\), representing the number of coins. Each of the following \(n\) lines contains two integers \(x_i\) and \(y_i\) \(( -10^9 \le x_i \le 10^9,\ 0 \le y_i \le 10^9 )\), denoting the initial coordinates of the \(i\)-th coin.

outputFormat

Output a single line: Abletocatch if FJ can catch all coins, or Notabletocatch otherwise.

sample

3
0 2
1 4
2 6

Abletocatch