
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 the planes \(S_{1}\) and \(S_{2}\) be perpendicular to each other, intersecting along a given half-line \(e\) with endpoint \(O\). In one of the four space quadrants defined by \(S_{1}\) and \(S_{2}\), two half-lines starting from \(O\) are given, which enclose an angle of \(60^{\circ}\) with \(e\) and have angles of \(30^{\circ}\) and \(45^{\circ}\) with \(S_{1}\). What is the angle between the plane defined by these two half-lines and the plane \(S_{1}\)?

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