
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
Consider a $3 \times 3$ matrix filled with distinct integers from 1 to 9 such that each row and each column contains numbers in strictly increasing order. Additionally, the sum of the numbers on the main diagonal (from top left to bottom right) is exactly 15. Determine the number of possible such matrices.

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