
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
Seven positive numbers form a geometric sequence \( a_{1}, a_{2}, \cdots, a_{7} \). The sum of the first five terms is \(\frac{62}{7 \sqrt{2}-6}\), and the sum of the last five terms is \(12 + 14 \sqrt{2}\). Then the product of these seven terms \( a_{1} a_{2} \cdots a_{7} \) is ________.

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