
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
In the movie ”Prison break  $4$ ”. Michael Scofield has to break into The Company. There, he encountered a kind of code to protect Scylla from being taken away.  This code require picking out every number in a  $2015\times 2015$  grid satisfying:
i) The number right above of this number is  $\equiv 1 \mod 2$ ii) The number right on the right of this number is     $\equiv 2 \mod 3$ iii) The number right below of this number is     $\equiv 3 \mod 4$ iv) The number right on the right of this number is     $\equiv 4 \mod 5$  .
How many number does Schofield have to choose?
Also,  in a  $n\times n$   grid,  the numbers from  $ 1$  to  $n^2$  are arranged in the following way :
On the first row, the numbers  are written in an ascending order  $1, 2, 3, 4, ..., n$ , each cell has one number.  
On the second row, the number are written in descending order  $2n, 2n -1, 2n- 2, ..., n + 1$ .   
On the third row,   it is ascending order again  $2n + 1, 2n + 2, ..., 3n$ .  
The numbers are written like that until  $n$ th row.  
For  example, this is how a  $3$   $\times$   $3$  board looks like:![Image](https://cdn.artofproblemsolving.com/attachments/8/7/0a5c8aba6543fd94fd24ae4b9a30ef8a32d3bd.png)

Present the answer in LaTex format: \boxed{Your answer}
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