
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
An textile factory's workshop has $n$ (where $n > 7$ and $n \in \mathbb{N}$) looms, numbered from 1 to $n$, and there are $n$ technical workers, also numbered from 1 to $n$. Define a notation $a_{ij}\ (1 \leq i, j \leq n)$ such that $a_{ij} = 1$ if worker $i$ operates loom $j$, and $a_{ij} = 0$ otherwise. Among the following propositions, identify all the correct statements:
① If loom number 7 is operated by exactly one person, then $a_{17}+a_{27}+a_{37}+\dots+a_{n7} = 1$;
② If $a_{11}+a_{12}+\dots+a_{1n}+a_{21}+a_{22}+\dots+a_{2n} = 2$, it means worker number 1 and worker number 2 operate one loom each;
③ If $a_{11}+a_{12}+\dots+a_{1n}+a_{21}+a_{22}+\dots+a_{2n} = 2$, it indicates that loom number 1 and loom number 2 are operated by two workers;
④ If $a_{31}+a_{32}+a_{33}+\dots+a_{3n} = 2$, it implies that worker number 3 operates two looms.

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