
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 \(n^2 (n \geqslant 4)\) positive numbers arranged in an \(n\) by \(n\) grid:
$$
\begin{array}{llllll}
a_{11} & a_{12} & a_{13} & a_{14} & \cdots & a_{1 n} \\
a_{21} & a_{2} & a_{23} & a_{24} & \cdots & a_{2 n} \\
a_{31} & a_{32} & a_{33} & a_{34} & \cdots & a_{3 n} \\
\cdots & & & & & \\
a_{n 1} & a_{n 2} & a_{n 3} & a_{n 4} & \cdots & a_{n n}
\end{array}
$$

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

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