
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
If a coin lands on the same side $k$ times in a row, we will call such a subsequence a group of length $k$. For example, in the sequence HHTHHTHH there are four groups, with lengths $1, 2, 2$ and $2$, respectively.

Denote $p_{n, k}$ as the probability that in a series of $n$ tosses of a symmetric coin, there is a group of length $k$ or more ($k < n$). Derive the recurrence relation:

$$
p_{n, k} = p_{n-1, k} - \frac{1}{2^{k}} p_{n-k, k} + \frac{1}{2^{k}}
$$

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