
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 \(n^{2}\) positive numbers arranged in \(n\) rows and \(n\) columns (where \(n \geq 4\) ):
\[ 
\begin{array}{l}
a_{11} a_{12} a_{13} a_{14} \cdots a_{1n} \\
a_{21} a_{22} a_{23} a_{24} \cdots a_{2n} \\
a_{31} a_{32} a_{33} a_{34} \cdots a_{3n} \\
\cdots \cdots \cdots \cdots \cdots \\
a_{n1} a_{n2} a_{n3} a_{n4} \cdots a_{nn}
\end{array}
\]

Each row forms an arithmetic sequence, each column forms a geometric sequence, and all columns share the same common ratio. Given that \( a_{24} = 1 \), \( a_{42} = \frac{1}{8} \), and \( a_{43} = \frac{3}{16} \), find the sum \( a_{11} + a_{22} + a_{23} + \cdots + a_{nn} \).

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