Given below is a well-thought out solution to a math problem generated by an AI system. The system consists of a planner and an executor. The planner model thinks out loud and plans the next component of the problem solution. Then, it provides a prompt along with the plan to an executor model. The executor then follows the instructions in the prompt and uses context from the plan to carry out the given task.

The solution is broken down into multiple components, each containing the following:
- Description: A brief description of what the component does. It is provided within <description> tags.
- Plan: The plan generated by the planner provided within <plan> tags.
- Prompt: The instructions generated by the planner provided within <prompt> tags.
- Execution: The output of the executor provided within <execution> tags.

Though the executions are run sequentially in this solution, some of the executions may be parallelized to improve speed. Identify and explain which components can run in parallel and determine the best way to parallelize them to maximize speed. Note that parallel runs should not have co-dependency.

The parallelization schedule can be represented as a directed acyclic graph (DAG) where the nodes are the component numbers. You need to represent the DAG as a parent dictionary where each node is a key and its value is a list of nodes that point to it, i.e., the nodes that must be executed immediately before it. For a key node, do not include any nodes in its value that can be run in parallel with it.

Format of parent dictionary:

Let us consider a simple example. Suppose that component 1 needs to be run before any other component. Then, components 2, 3, 4 can be run in parallel after 1. Component 5 which depends on the results of 2 and 3 can be run after 2 and 3. Component 6 which depends on the results of 4 and 5 can be run after 4 and 5. The parent dictionary for this example *MUST* be represented as a python dictionary as follows:
```python
parent_dictionary = {
    1: [],
    2: [1],
    3: [1],
    4: [1],
    5: [2, 3],
    6: [4, 5]
}
```