
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
In the cube $ABCD-A_1B_1C_1D_1$, points $M$ and $N$ are on $AB_1$ and $BC_1$ respectively (where $M$ and $N$ do not coincide with $B_1$ and $C_1$), and $AM=BN$. Then, among the following statements: ① $AA_1 \perp MN$; ② $A_1C_1 \parallel MN$; ③ $MN \parallel$ plane $A_1B_1C_1D_1$; ④ $MN$ and $A_1C_1$ are skew lines; the correct statements are \_\_\_\_\_\_\_\_.

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