
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) \( P \) contains both positive and negative elements;
(2) \( P \) contains both odd and even elements;
(3) \( 1 \notin P \);
(4) If \( x, y \in P \), then \( x + y \in P \).

Determine the relationship of 0 and 2 with the set \( P \).

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