
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
Below is a "right triangular number array":
\[
\frac{1}{4},
\]
\[
\frac{1}{2}, \quad \frac{1}{4}
\]
\[
\frac{3}{4}, \quad \frac{3}{8}, \quad \frac{3}{16}
\]
Each column forms an arithmetic sequence, starting from the third row, each row forms a geometric sequence with a constant common ratio. Let the number in the \(i^\text{th}\) row and \(j^\text{th}\) column be \(a_{i j}\) (\(i \geqslant j, i, j \in \mathbf{N}_{+}\)), find \(a_{n 3}\).

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