
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
Arrange 25 numbers in a 5 by 5 grid:

\(a\_{11}\)  \(a\_{12}\)  \(a\_{13}\)  \(a\_{14}\)  \(a\_{15}\)

\(a\_{21}\)  \(a\_{22}\)  \(a\_{23}\)  \(a\_{24}\)  \(a\_{25}\)

\(a\_{31}\)  \(a\_{32}\)  \(a\_{33}\)  \(a\_{34}\)  \(a\_{35}\)

\(a\_{41}\)  \(a\_{42}\)  \(a\_{43}\)  \(a\_{44}\)  \(a\_{45}\)

\(a\_{51}\)  \(a\_{52}\)  \(a\_{53}\)  \(a\_{54}\)  \(a\_{55}\)

The first row forms an arithmetic sequence, and each column forms a geometric sequence with the same common ratio for all columns. If \(a\_{24}=4\), \(a\_{41}=-2\), \(a\_{43}=10\), find the value of \(a\_{11}×a\_{55}\).

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