
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
Consider a sequence  $a_0$ ,  $a_1$ ,  $\ldots$ ,  $a_9$  of distinct positive integers such that  $a_0=1$ ,  $a_i < 512$  for all  $i$ , and for every  $1 \le k \le 9$  there exists  $0 \le m \le k-1$  such that \[(a_k-2a_m)(a_k-2a_m-1) = 0.\] Let  $N$  be the number of these sequences. Find the remainder when  $N$  is divided by  $1000$ .

*Based on a proposal by Gyumin Roh*

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