
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
When a non-empty set of numbers \( F \) satisfies the conditions: "If \( a, b \in F \), then \( a+b, a-b, ab \in F \), and if \( b \neq 0 \), then \( \frac{a}{b} \in F \)," \( F \) is called a field. Among the following four propositions about fields:
(1) 0 is an element of any field;
(2) If a field \( F \) has non-zero elements, then 2016 is in \( F \);
(3) The set \( P = \{x \mid x = 3k, k \in \mathbf{Z}\} \) is a field;
(4) The set of rational numbers is a field.
Which of the following propositions are true? (Write down all the numbers of the true propositions).

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