
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 a set $P \subseteq \mathbb{Z}$, and it satisfies the following conditions:
(1) There are positive numbers and negative numbers in $P$; (2) There are odd numbers and even numbers in $P$; (3) $-1 \notin P$; (4) If $x, y \in P$, then $x + y \in P$.
Now consider the following judgments: ① $1 \in P$; ② $1 \notin P$; ③ $2 \in P$; ④ $2 \notin P$; ⑤ $0 \in P$.
The correct judgments among these are numbered as .

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