
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  $N$  denote the number of ordered pairs of sets  $(A, B)$  such that  $A \cup B$  is a size- $999$  subset of  $\{1,2,\dots,1997\}$  and  $(A \cap B) \cap \{1,2\} = \{1\}$ . If  $m$  and  $k$  are integers such that  $3^m5^k$  divides  $N$ , compute the the largest possible value of  $m+k$ .

*Proposed by Michael Tang*

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