
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
450. Signal lights. Two spies on opposite sides of a river devised a method of nighttime signaling using a frame (similar to the one shown in the image) and three lamps. Each lamp could shine white, red, or green light. The spies developed a code in which each signal meant something specific.

You understand, of course, that one lamp, no matter which hook it is hung on, will always have the same value. Two lamps hanging on the top hooks 1 and 2 are indistinguishable from two lamps hanging on hooks 4 and 5. Two red lamps on hooks 1 and 5 can be distinguished from lamps on hooks 1 and 6, but two lamps on hooks 1 and 2 can be distinguished from two lamps on hooks 1 and 3.

Considering all these different positions of the lamps on the hooks and the colors of the signals, how many different signals can be sent?

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