
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 two non-zero vectors $\mathbf{a}$ and $\mathbf{b}$ that are perpendicular to each other, consider the following statements:
1. $\mathbf{a} \cdot \mathbf{b} = 0$
2. $\mathbf{a} + \mathbf{b} = \mathbf{a} - \mathbf{b}$
3. $|\mathbf{a} + \mathbf{b}| = |\mathbf{a} - \mathbf{b}|$
4. $|\mathbf{a}|^2 + |\mathbf{b}|^2 = (\mathbf{a} + \mathbf{b})^2$
5. $(\mathbf{a} + \mathbf{b}) \cdot (\mathbf{a} - \mathbf{b}) = 0$  
Identify all the correct statements from the list above.

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