
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  $0 \le m,n \le 64$ , 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  $65 \times 65$  matrix  $M$  whose  $(i,j)$ th entry (for  $1 \le i, j \le 65$ ) is \[ (-1)^{\alpha(i-1, j-1)}. \] Compute the remainder when  $\det M$  is divided by  $1000$ .

*Proposed by Evan Chen*

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