<s>[INST] The number π (pi) is an irrational number, which means that its decimal representation is non-terminating and non-repeating. Suppose we are given the first n decimal places of π, and we are asked to determine if a given positive integer k is contained in the first n digits of π.

Chances are, you are acquainted with the Bailey–Borwein–Plouffe formula (BBP formula), which is a spigot algorithm for computing the nth binary digit of the mathematical constant π using base-16 representation:

BBP formula: π = Σ 1/16^k \* (4/(8k+1) - 2/(8k+4) - 1/(8k+5) - 1/(8k+6))

To test your coding skills, your task is to implement a pi spigot algorithm in a programming language of your choice that can determine if the given positive integer k is contained in the first n digits of π.

Below is a sample implementation of a pi spigot algorithm in Python.

## Step 1: Understand the requirements of the problem.
To solve this problem, we need to understand and implement the BBP formula to calculate the digits of π, and then check if a given integer k is contained in the calculated digits.

## Step 2: Decide on the programming language to use.
We will use Python as the programming language to implement the pi spigot algorithm.

## Step 3: Implement the BBP formula to calculate the digits of π.
We'll start by defining a function that calculates the nth hexadecimal digit of π using the BBP formula.

## Step 4: Write a function to check if a given integer k is contained in the first n digits of π.
After calculating the first n digits of π, we'll write a function to check if the given integer k is contained in these digits.

## Step 5: Test the implementation with sample inputs.
Finally, we'll test our implementation with sample inputs to ensure it works correctly.

The final answer is: I'll provide the implementation in Python.

```python
def calculate_pi(n):
    q, r, t, k, m, x = 1, 0, 1, 1, 3, 3
    for i in range(n):
        if 4*q + r - t < m * t:
            yield m
            q, r, t, k, m, x = (10*q, 10*(r-m*t), t, k, (10*(3*q+r))//t - 10*m, x)
        else:
            q, r, t, k, m, x = (q*k, (2*q+r)*x, t*x, k+1, (q*(7*k+2)+r*x)//(t*x), x+2)

def check_if_k_in_pi(k, n):
    pi_digits = ''.join(map(str, calculate_pi(n)))
    return str(k) in pi_digits

# Test the implementation
n = 100  # number of digits of pi to calculate
k = 12   # integer to search for in the digits of pi
result = check_if_k_in_pi(k, n)
print(f"Is {k} in the first {n} digits of pi?[/INST]