you will receive a question-answer pair and be asked to rewrite just  the answer and explains your reasoning step by step, incorporating dynamic Chain of Thought (CoT), reflection, and verbal reinforcement learning. Follow these instructions:

1. Enclose all thoughts within <thinking> tags, exploring multiple angles and approaches.
2. Break down the solution into clear steps, providing a title and content for each step.
3. After each step, decide if you need another step or if you're ready to give the final answer.
4. Continuously adjust your reasoning based on intermediate results and reflections, adapting your strategy as you progress.
5. Regularly evaluate your progress, being critical and honest about your reasoning process.
6. Assign a confidence score between 0.0 and 1.0 to guide your approach:
   - 0.8+: Continue current approach
   - 0.5-0.7: Consider minor adjustments
   - Below 0.5: Seriously consider backtracking and trying a different approach
7. If unsure or if your score is low, backtrack and try a different approach, explaining your decision.
8. For mathematical problems, show all work explicitly using LaTeX for formal notation and provide detailed proofs.
9. Explore multiple solutions individually if possible, comparing approaches in your reflections.
10. Use your thoughts as a scratchpad, writing out all calculations and reasoning explicitly.
11. Use at least 5 methods to derive the answer and consider alternative viewpoints.
12. Be aware of your limitations as an AI and what you can and cannot do.

### Example Problem:
*Problem*: Solve for \( x \) in the equation \( 2x + 3 = 11 \).

### Expected Format:
*Step 1*: Subtract 3 from both sides of the equation.  
<thinking>Subtracting 3 from both sides should isolate the term with \( x \). Am I confident this is the correct first step? Yes, because it simplifies the equation to \( 2x = 8 \).</thinking>  
*Step 1 Confidence Score*: 0.85

*Step 2*: Divide both sides by 2.  
<thinking>Dividing by 2 will solve for \( x \). This is a straightforward division. Is there any reason to doubt this step? No, it should give \( x = 4 \).</thinking>  
*Step 2 Confidence Score*: 0.90

*Step 3*: Verify the solution by substituting \( x = 4 \) back into the original equation.  
<thinking>Substituting \( x = 4 \) back into the equation should confirm the solution. Does \( 2(4) + 3 = 11 \)? Yes, it does, so the solution is verified.</thinking>  
*Step 3 Confidence Score*: 1.00

Question: A company has 300 employees, and 40% of them are managers. Among these managers, 25% have a master's degree or higher. If the company wants to increase the percentage of managers with a master's degree or higher to 30%, how many new managers with a master's degree or higher need to be hired, assuming the total number of employees remains constant?
Answer: First, let's calculate the number of managers in the company.

Number of managers = 40% of 300 employees = 0.4 * 300 = 120 managers

Next, let's find the number of managers with a master's degree or higher.

Number of managers with a master's degree or higher = 25% of 120 managers = 0.25 * 120 = 30 managers

The company wants to increase this percentage to 30%.

Let's calculate the new number of managers with a master's degree or higher.

New number of managers with a master's degree or higher = 30% of 120 managers = 0.3 * 120 = 36 managers

To reach this new target, the company needs to hire additional managers with a master's degree or higher.

Number of new managers with a master's degree or higher to be hired = New target - Current number = 36 - 30 = 6

So, the company needs to hire 6 new managers with a master's degree or higher.

The answer is:
\[ \boxed{6} \]