"""Create reasoning_engine
Generated by Phi-Octopus Eden
2025-11-06 04:14:56.068548
"""

```python
from typing import List, Dict, Tuple

class ReasoningEngine:
    """
    A simple reasoning engine that solves a limited reasoning problem.
    
    This engine is designed to identify patterns in a sequence of numbers and
    predict the next number based on a linear relationship found within the input data.
    """

    def __init__(self):
        self.pattern = None

    def train(self, data: List[int]) -> None:
        """
        Train the engine with a list of integers to identify a linear pattern.

        :param data: A list of integers representing a sequence.
        """
        if len(data) < 2:
            raise ValueError("At least two points are required for training.")
        
        x, y = range(len(data)), data
        m, b = self.calculate_linear_fit(x, y)
        self.pattern = (m, b)

    def calculate_linear_fit(self, x: List[int], y: List[int]) -> Tuple[float, float]:
        """
        Calculate the linear fit coefficients for a set of points.

        :param x: A list of independent variable values.
        :param y: A list of dependent variable values corresponding to `x`.
        :return: The slope and intercept of the best-fit line.
        """
        n = len(x)
        sum_x, sum_y = sum(x), sum(y)
        mean_x, mean_y = sum_x / n, sum_y / n
        ss_xx = sum((xi - mean_x) ** 2 for xi in x)
        ss_xy = sum((xi - mean_x) * (yi - mean_y) for xi, yi in zip(x, y))
        
        m = ss_xy / ss_xx
        b = mean_y - m * mean_x
        
        return m, b

    def predict(self, next_index: int) -> float:
        """
        Predict the value at a given index based on the learned pattern.

        :param next_index: The index for which to predict the value.
        :return: The predicted value.
        """
        if self.pattern is None:
            raise ValueError("Engine must be trained before making predictions.")
        
        m, b = self.pattern
        return m * next_index + b

# Example usage:
if __name__ == "__main__":
    engine = ReasoningEngine()
    training_data = [1, 2, 3, 4, 5]
    engine.train(training_data)
    
    print(f"Predicted value for index 6: {engine.predict(6)}")
```