
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
The vertices of a regular tetrahedron are labeled \(1, 2, 3\), and \(4\). Reflect an arbitrary point \(P\) in vertex \(1\), yielding reflection \(P_{1}\). Then reflect \(P_{1}\) in vertex \(2\), yielding reflection \(P_{12}\). Next, reflect \(P_{12}\) in vertex \(3\), yielding reflection \(P_{123}\). Finally, reflect \(P_{123}\) in vertex \(4\), yielding reflection \(P_{1234}\). How can the result of the four reflections be obtained more simply? How does the result change with the 24 different permutations of the reflections?

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