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: Tom has a collection of coins from different countries. The coins are made of different metals: copper, silver, and gold. Tom wants to put the coins into small pouches, with each pouch containing coins from only one country and made of only one metal. The pouches can hold a maximum of 5 coins each. Tom has the following coins:

- 10 copper coins from the USA, each worth $0.05
- 8 silver coins from Canada, each worth $0.10
- 12 gold coins from Mexico, each worth $0.20
- 5 copper coins from the UK, each worth $0.05
- 6 silver coins from Australia, each worth $0.10

How many pouches can Tom fill with the coins, and what is the total value of the coins in each pouch?
Answer: To solve the problem, we need to calculate the number of pouches Tom can fill with the coins from each country and made of each metal.

1. Copper coins from the USA:
   \[ \text{Number of pouches} = \text{Total coins} : 5 = 10 : 5 = 2 \]
   \[ \text{Total value per pouch} = 5 \times \$0.05 = \$0.25 \]

2. Silver coins from Canada:
   \[ \text{Number of pouches} = \text{Total coins} : 5 = 8 : 5 = 1 \text{ with } 3 \text{ coins left over} \]
   \[ \text{Total value per pouch} = 5 \times \$0.10 = \$0.50 \]
   For the 3 leftover coins, Tom can either keep them separate or combine them with coins from another country made of the same metal (if possible).

3. Gold coins from Mexico:
   \[ \text{Number of pouches} = \text{Total coins} : 5 = 12 : 5 = 2 \text{ with } 2 \text{ coins left over} \]
   \[ \text{Total value per pouch} = 5 \times \$0.20 = \$1.00 \]
   For the 2 leftover coins, Tom can either keep them separate or combine them with coins from another country made of the same metal (if possible).

4. Copper coins from the UK:
   \[ \text{Number of pouches} = \text{Total coins} : 5 = 5 : 5 = 1 \]
   \[ \text{Total value per pouch} = 5 \times \$0.05 = \$0.25 \]

5. Silver coins from Australia:
   \[ \text{Number of pouches} = \text{Total coins} : 5 = 6 : 5 = 1 \text{ with } 1 \text{ coin left over} \]
   \[ \text{Total value per pouch} = 5 \times \$0.10 = \$0.50 \]
   For the 1 leftover coin, Tom can either keep it separate or combine it with coins from another country made of the same metal (if possible).

Tom can fill a total of $2 + 1 + 2 + 1 + 1 = 3 + 2 + 1 + 1 = 5 + 1 + 1 = 6 + 1 = 7$ pouches.

As for the leftover coins, Tom has the following options:

- 3 silver coins from Canada
- 2 gold coins from Mexico
- 1 silver coin from Australia

Since Tom wants each pouch to contain coins from only one country and made of only one metal, he cannot combine the leftover coins with coins from other countries.

Tom can keep the leftover coins separate or consider alternative storage options.

The answer is:
\[ \boxed{7} \]