
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
Mark writes the expression \(\sqrt{d}\) for each positive divisor \(d\) of \(8!\) on the board. Rishabh then simplifies each expression to the form \(a \sqrt{b}\), where \(a\) and \(b\) are integers, and \(b\) is not divisible by the square of a prime number. For example, \(\sqrt{20}, \sqrt{16}\), and \(\sqrt{6}\) simplify to \(2 \sqrt{5}, 4 \sqrt{1}\), and \(1 \sqrt{6}\), respectively. Compute the sum of \(a + b\) across all expressions that Rishabh writes.

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