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 sells two types of smartphones: a basic model and a premium model. The basic model costs $\$300$, and the premium model costs $\$600$. According to the company's marketing research, the number of customers who buy the basic model is directly proportional to the price difference between the two models, while the number of customers who buy the premium model is inversely proportional to the price difference. If 150 customers buy the basic model and 50 customers buy the premium model, what is the expected number of customers who would buy the basic model if the price of the premium model were increased to $\$900$?
Answer: ## Step 1: Define the variables and the relationships between them.
Let's denote the price of the basic model as $B = 300$, the original price of the premium model as $P_1 = 600$, and the new price of the premium model as $P_2 = 900$. The price difference between the basic and the premium models originally is $D_1 = P_1 - B$, and after the price increase of the premium model, it becomes $D_2 = P_2 - B$.

## Step 2: Determine the relationship between the number of customers and the price difference for each model.
Since the number of customers who buy the basic model is directly proportional to the price difference between the two models, we can write this relationship as $N_{basic} = k \cdot D$, where $k$ is the constant of proportionality and $D$ is the price difference.

For the premium model, the number of customers is inversely proportional to the price difference, which can be written as $N_{premium} = m / D$, where $m$ is another constant of proportionality.

## Step 3: Use the given data to find the constants of proportionality.
We are given that when the price difference is $D_1 = 600 - 300 = 300$, the number of customers for the basic model is $N_{basic} = 150$ and for the premium model is $N_{premium} = 50$.

For the basic model, we have $150 = k \cdot 300$, so $k = 150 / 300 = 0.5$.

For the premium model, we have $50 = m / 300$, so $m = 50 \cdot 300 = 15000$.

## Step 4: Calculate the expected number of customers for the basic model after the price increase.
The new price difference after increasing the price of the premium model to $900$ is $D_2 = 900 - 300 = 600$.

Using the direct proportionality relationship for the basic model with the new price difference, we find $N_{basic\_new} = 0.5 \cdot 600 = 300$.

The final answer is: $\boxed{300}$