
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
Let \( a, b \in \mathbb{Z} \) and \( m \) be a positive integer. If the remainders of \( a \) and \( b \) when divided by \( m \) are the same, then \( a \) and \( b \) are said to be congruent modulo \( m \), denoted as \( a \equiv b \pmod{m} \). Among the following statements:
(1) If \( a \equiv b \pmod{m} \) and \( d \) is a positive divisor of \( m \), then \( a \equiv b \pmod{d} \);
(2) If \( a \equiv b \pmod{m} \) and \( a \equiv b \pmod{n} \), then \( a \equiv b \pmod{mn} \);
(3) If \( c \) is a positive integer and \( ac \equiv bc \pmod{m} \), then \( a \equiv b \pmod{m} \);
(4) Let \( f(x)=x^{3}-2x+5 \). If \( a \equiv b \pmod{m} \), then \( f(a) \equiv f(b) \pmod{m} \).

The correct propositions are (write down all correct proposition numbers).

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