
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
There are \( n^2 \) positive numbers arranged in \( n \) rows and \( n \) columns (\( n \geq 4 \)):
\[
\begin{array}{llllll}
a_{11} & a_{12} & a_{13} & a_{14} & \cdots & a_{1n} \\
a_{21} & a_{22} & a_{23} & a_{24} & \cdots & a_{2n} \\
a_{31} & a_{32} & a_{33} & a_{34} & \cdots & a_{3n} \\
a_{41} & a_{42} & a_{43} & a_{44} & \cdots & a_{4n} \\
\cdots & \cdots 
\end{array}
\]

Each row forms an arithmetic sequence, each column forms a geometric sequence, and all columns have the same common ratio.

Given \( a_{24}=1 \), \( a_{42}=\frac{1}{8} \), and \( a_{43}=\frac{3}{16} \), find
\[
a_{11} + a_{22} + a_{33} + a_{44} + \cdots + a_{nn}.
\]

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