
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
In an infinite chain of nerve cells, each can be in one of two states: "rest" and "excitation". If at a given moment a cell is excited, it sends a signal that reaches both its neighboring cells in one unit of time (say, one millisecond). Each cell becomes excited if and only if it receives a signal from one of its neighboring cells; if signals come simultaneously from both sides, they cancel each other, and the cell does not become excited. For example, if at the initial moment of time \( t = 0 \) three neighboring cells are excited, while the others remain at rest, the excitation will propagate as shown in the illustration.

Suppose at the initial moment of time only one cell is excited. How many cells will be in the excited state after 15 milliseconds? After 65 milliseconds? After 1000 milliseconds? In general, after \( t \) milliseconds?

What happens if the chain is not infinite but consists of \( N \) cells connected in a circle? Will the excitation be sustained indefinitely or will it die out?

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