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: In a botanical garden, there are three types of flowers: roses, daisies, and lilies. The ratio of the number of roses to the number of daisies is 4:5, and the ratio of the number of daisies to the number of lilies is 3:2. If there are 120 flowers in total, how many lilies are there?
Answer: Let's use the given ratios to find the number of lilies.

First, we need to find a common ratio for all three types of flowers. The given ratios are:

Roses to daisies: 4:5
Daisies to lilies: 3:2

To find the common ratio, we can multiply the two ratios together, making sure to match the middle term (daisies):
Roses : daisies = 4 : 5
Daisies : lilies = 3 : 2

First, we find the least common multiple of the two different values for daisies (5 and 3), which is 15. Then we can rewrite the ratios as:
Roses : daisies = 4*3 : 5*3 = 12 : 15
Daisies : lilies = 3*5 : 2*5 = 15 : 10

Now we have the common ratio for all three types of flowers: 12:15:10.

Let's say the number of roses is 12x, the number of daisies is 15x, and the number of lilies is 10x.

We know that the total number of flowers is 120, so we can set up the equation:
12x + 15x + 10x = 120

Combine like terms:
37x = 120

Divide by 37:
x = 120 : 37

Now that we have the value of x, we can find the number of lilies:
Number of lilies = 10x = 10(120 : 37) = 1200 : 37

Simplify the fraction:
\[ 1200 : 37 = 32.43 \approx 32 \]

Since we can't have a fraction of a flower, we'll round down to the nearest whole number.

So, there are 32 lilies in the botanical garden.

Answer:
\[ \boxed{32} \]