
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
Adva van egy \( AB = a \) távolság. Szerkesszük meg e távolságon a \( P \) pontot úgy, hogy az \( AP \), illetőleg \( PB \) távolságokkal, mint élekkel alkotott szabályos tetraéder, illetőleg szabályos hexaéder felületeinek összege a lehető legkisebb legyen.

 (English Translation)
Given a distance \( AB = a \). Construct a point \( P \) on this distance such that the sum of the surface areas of a regular tetrahedron and a regular hexahedron, formed with \( AP \) and \( PB \) as edges respectively, is minimized.

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