
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
For integers  $0 \le m,n \le 2^{2017}-1$ , let  $\alpha(m,n)$  be the number of nonnegative integers  $k$  for which  $\left\lfloor m/2^k \right\rfloor$  and  $\left\lfloor n/2^k \right\rfloor$  are both odd integers.  Consider a  $2^{2017} \times 2^{2017}$  matrix  $M$  whose  $(i,j)$ th entry (for  $1 \le i, j \le 2^{2017}$ ) is \[ (-1)^{\alpha(i-1, j-1)}. \] For  $1 \le i, j \le 2^{2017}$ , let  $M_{i,j}$  be the matrix with the same entries as  $M$  except for the  $(i,j)$ th entry, denoted by  $a_{i,j}$ , and such that  $\det M_{i,j}=0$ . Suppose that  $A$  is the  $2^{2017} \times 2^{2017}$  matrix whose  $(i,j)$ th entry is  $a_{i,j}$  for all  $1 \le i, j \le 2^{2017}$ . Compute the remainder when  $\det A$  is divided by  $2017$ .

*Proposed by Michael Ren and Ashwin Sah*

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