
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
Define a new operation between two real numbers "\*" as follows: $x*y=\lg(10^x+10^y)$, where $x, y \in \mathbb{R}$. For any real numbers $a$, $b$, $c$, consider the following statements: 
1. $a*b=b*a$
2. $(a*b)*c=a*(b*c)$
3. $(a*b)+c=(a+c)*(b+c)$
4. $(a*b) \times c=(a \times c)*(b \times c)$

Among these statements, the correct ones are:

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