
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  $ X$  the set of all sequences  $ \{a_1, a_2,\ldots , a_{2000}\}$ , such that each of the first 1000 terms is 0, 1 or 2, and each of the remaining terms is 0 or 1. The *distance* between two members  $ a$  and  $ b$  of  $ X$  is defined as the number of  $ i$  for which  $ a_i$  and  $ b_i$  are different.

Find the number of functions  $ f : X \to X$  which preserve the distance.

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