
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
Read the following material: "Why is $\sqrt[3]{2}$ not a rational number" and complete the question.  
Proof: Assume that $\sqrt[3]{2}$ is a rational number.  
Then there exist two coprime positive integers $n$ and $m$ such that $\sqrt[3]{2} = \frac{n}{m}$, then ____.  
Since $n^{3}$ is a multiple of $2$,  
Therefore ____.  
Let $n = 2t$ (where $t$ is a positive integer), then $n^{3} = 8t^{3}$,  
Therefore ____, that is $4t^{3} = m^{3}$,  
Therefore ____,  
Therefore $m$ and $n$ are both multiples of $2$, not coprime, which contradicts the assumption.  
Therefore the assumption is not valid, so $\sqrt[3]{2}$ is not a rational number.  
Fill in the blanks in the material with the following options in order.  
① $8t^{3} = 2m^{3}$; ② $n^{3} = 2m^{3}$; ③ $m$ is a multiple of $2$; ④ $n$ is a multiple of $2$.

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