
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
On the sides \(AB\) and \(AC\) of the acute-angled triangle \(ABC\), two equal rectangles \(AMNB\) and \(APQC\) are constructed outwardly. Find the distance between the vertices \(N\) and \(Q\) of the rectangles if the lengths of the sides \(AB\) and \(AC\) are \(2\sqrt{2}\) and 1, respectively, and the angle at vertex \(A\) of the triangle is \(45^\circ\).

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