
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 vectors $\mathbf{a} = (m, n)$ and $\mathbf{b} = (p, q)$, define $\mathbf{a} \otimes \mathbf{b} = mn - pq$. Consider the following four statements: 
1. $\mathbf{a} \otimes \mathbf{a} = 0$;
2. $\mathbf{a} \otimes \mathbf{b} = \mathbf{b} \otimes \mathbf{a}$;
3. $(\mathbf{a} + \mathbf{b}) \otimes \mathbf{a} = \mathbf{a} \otimes \mathbf{a} + \mathbf{b} \otimes \mathbf{a}$;
4. $(\mathbf{a} \otimes \mathbf{b})^2 + (\mathbf{a} \cdot \mathbf{b})^2 = (m^2 + q^2) \cdot (n^2 + p^2)$.

Which of the statements are correct? Write down the numbers of all correct statements.

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