
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
All dihedral angles of the trihedral angle \(N K L M\) (where \(N\) is the vertex) are right angles. A point \(P\) is taken on the face \(L N M\) at a distance of 2 from the vertex \(N\) and a distance of 1 from the edge \(M N\). From a point \(S\) located inside the trihedral angle \(N K L M\), a beam of light is directed towards point \(P\). The beam forms an angle of \(\pi / 4\) with the plane \(M N K\) and equal angles with the edges \(K N\) and \(M N\). The beam is reflected off the faces of the angle \(N K L M\) first at point \(P\), then at point \(Q\), and then at point \(R\). Find the sum of the lengths of segments \(P Q\) and \(Q R\).

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