
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
Given the set $P$ consisting of positive integers not exceeding 2012, explicitly $P = \{1,2,3, \cdots, 2012\}$. The set $A$ is a subset of $P$, where $|A|$ denotes the number of elements in set $A$, and $S(A)$ denotes the sum of all elements in set $A$.

1. If the difference between any two numbers in set $A$ is not a multiple of 101, find the maximum value of $|A|$.
2. If the difference between any two numbers in set $A$ is not a multiple of 101, and the sum of any two numbers is also not a multiple of 101, find the maximum value of $|A|$.
3. If the difference between any two numbers in set $A$ is not a multiple of 101, the sum of any two numbers is also not a multiple of 101, and $S(A) = 2012$, find the maximum value of $|A|$.

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