
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
A game consists in pushing a flat stone along a sequence of squares  $S_0, S_1, S_2, . . .$  that are arranged in linear order. The stone is initially placed on square  $S_0$ . When the stone stops on a square  $S_k$  it is pushed again in the  same direction and so on until it reaches  $S_{1987}$  or goes beyond it; then the game stops. Each time the stone is pushed, the probability that it will advance exactly  $n$  squares is  $\frac{1}{2^n}$ . Determine the probability that the stone will stop exactly on square  $S_{1987}.$ 

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