
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
(i) Consider two positive integers  $a$  and  $b$  which are such that  $a^a b^b$  is divisible by  $2000$ . What is the least possible value of  $ab$ ?
(ii) Consider two positive integers  $a$  and  $b$  which are such that  $a^b b^a$  is divisible by  $2000$ . What is the least possible value of  $ab$ ?

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