
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
A set \( P \) of certain integers has the following properties:  
(1) The set \( P \) contains both positive and negative elements.  
(2) The set \( P \) contains both odd and even elements.  
(3) \(1 \notin P \).  
(4) If \( x, y \in P \), then \( x + y \in P \).  

Determine the relation between the elements 0, 2, and the set \( P \).

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