
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 \( a+b, a-b, a \cdot b \in F \), and when \( b \neq 0 \), \( \frac{a}{b} \in F \)", then \( F \) is called a field. Given the following four propositions about fields:
(1) 0 is an element of any field;
(2) If a field \( F \) has a non-zero element, then 2016 is in \( F \);
(3) The set \( P=\{x \mid x=3 k, k \in \mathbf{Z}\} \) is a field;
(4) The set of rational numbers is a field.

Identify the true propositions.

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