
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

$n^{2}$ positive numbers are arranged in $n$ rows and $n$ columns ($n \geq 4$),
$$
\begin{array}{llllll}
a_{11} & a_{12} & a_{13} & a_{14} & \cdots & a_{1n} \\
a_{21} & a_{22} & a_{23} & a_{24} & \cdots & a_{2n} \\
a_{31} & a_{32} & a_{33} & a_{34} & \cdots & a_{3n} \\
a_{41} & a_{42} & a_{43} & a_{44} & \cdots & a_{4n} \\
\cdots & & & & & \\
a_{n1} & a_{n2} & a_{n3} & a_{n4} & \cdots & a_{nn}
\end{array}
$$

In each row, the numbers form an arithmetic sequence, and in each column, the numbers form a geometric sequence with the same common ratio. Given that $a_{24} = 1$, $a_{42} = \frac{1}{8}$, 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}
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