
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 matrix composed of 9 distinct positive numbers $$\begin{pmatrix} \left.\begin{matrix}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{matrix}\right.\end{pmatrix}$$, the three numbers in each row form an arithmetic sequence, and $a_{11}+a_{12}+a_{13}$, $a_{21}+a_{22}+a_{23}$, $a_{31}+a_{32}+a_{33}$ form a geometric sequence. The correct number of the following four judgments is \_\_\_\_\_\_.  
① The second column $a_{12}$, $a_{22}$, $a_{32}$ must form a geometric sequence.  
② The first column $a_{11}$, $a_{21}$, $a_{31}$ do not necessarily form a geometric sequence.  
③ $a_{12}+a_{32} > a_{21}+a_{23}$.  
④ If the sum of the 9 numbers equals 9, then $a_{22} < 1$.

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