
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  $p$  be a fixed odd prime. Find the minimum positive value of  $E_{p}(x,y) = \sqrt{2p}-\sqrt{x}-\sqrt{y}$  where  $x,y \in \mathbb{Z}_{+}$ .

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