
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 with \( n \geq 4 \):

\[
\begin{array}{llllll}
a_{11} & a_{12} & a_{13} & a_{14} & \cdots & a_{1n} \\
a_{21} & a_{22} & a_{23} & a_{24} & \cdots & a_{2n} \\
a_{31} & a_{32} & a_{33} & a_{34} & \cdots & a_{3n} \\
\cdots & & & & & \\
a_{n1} & a_{n2} & a_{n3} & a_{n4} & \cdots & a_{nn}
\end{array}
\]

In which each row is an arithmetic progression (i.e., the numbers in each row form an arithmetic sequence), each column is a geometric progression (i.e., the numbers in each column form a geometric sequence), and the common ratio of all geometric progressions is the same. Given that \( a_{24}=1 \), \( a_{42}=\frac{1}{8} \), and \( a_{43}=\frac{3}{16} \), find the sum \( a_{11}+a_{22}+\cdots+a_{nn} \).

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