
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 a positive integer. A pair of  $n$ -tuples  $(a_1,\cdots{}, a_n)$  and  $(b_1,\cdots{}, b_n)$  with integer entries is called an *exquisite pair* if  $$ |a_1b_1+\cdots{}+a_nb_n|\le 1. $$  Determine the maximum number of distinct  $n$ -tuples with integer entries such that any two of them form an exquisite pair.

*Pakawut Jiradilok and Warut Suksompong, Thailand*

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