
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
Let  $a, b, c, d$  be an increasing arithmetic sequence of positive real numbers with common difference  $\sqrt2$ . Given that the product  $abcd = 2021$ ,  $d$  can be written as  $\frac{m+\sqrt{n}}{\sqrt{p}}$  , where  $m, n,$  and  $p$  are positive integers not divisible by the square of any prime. Find  $m + n + p$ .

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