
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
Given an array of $n^2$ positive numbers arranged into $n$ rows and $n$ columns (where $n \geq 4$ and $n \in \mathbb{N}^*$):  
            Column 1     Column 2    Column 3   …Column n  
Row 1        $a_{1,1}$ $a_{1,2}$ $a_{1,3}$ …$a_{1,n}$  
Row 2        $a_{2,1}$ $a_{2,2}$ $a_{2,3}$ …$a_{2,n}$  
Row 3         $a_{3,1}$ $a_{3,2}$ $a_{3,3}$ …$a_{3,n}$  
…  
Row n         $a_{n,1}$ $a_{n,2}$ $a_{n,3}$ …$a_{n,n}$  
where $a_{i,k}$ ($i, k \in \mathbb{N}^*$, and $1 \leq i \leq n$, $1 \leq k \leq n$) represents the number in the $i$-th row and $k$-th column of the array. It is known that the numbers in each row form an arithmetic sequence, and the numbers in each column form a geometric sequence with a common ratio of 2. Given that $a_{2,3} = 8$, $a_{3,4} = 20$, find the value of $a_{2,2}$.

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