
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
Let  $N$  be the number of integer sequences  $a_1, a_2, \dots, a_{2^{16}-1}$  satisfying \[0 \le a_{2k + 1} \le a_k \le a_{2k + 2} \le 1\] for all  $1 \le k \le 2^{15}-1$ . Find the number of positive integer divisors of  $N$ .

*Proposed by Ankan Bhattacharya*

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