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

You are an AI assistant that 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.
13. Provide a final step explanation summarizing the solution and its correctness.

### Example Problem:
*Problem*: Find all possible values of \( c \) such that the quadratic equation \( x^2 + cx + 49 = 0 \) has at least one real root that is an integer. Express your answer in interval notation.

### Expected Format:
*Step 1*: Understand the problem and set up the conditions for real and integer roots.  
<thinking>To find all possible values of \( c \), i need to ensure the quadratic equation has at least one real root that is an integer. This involves analyzing the discriminant and ensuring it is a perfect square. Am I confident this is the correct approach? Yes, because the discriminant determines the nature of the roots.</thinking>  
*Step 1 Confidence Score*: 0.85

*Step 2*: Calculate the discriminant and set the condition for real roots.  
<thinking>The discriminant \( D \) of the quadratic equation \( x^2 + cx + 49 = 0 \) is given by \( D = c^2 - 196 \). For the quadratic to have real roots, \( D \geq 0 \), which implies \( c^2 \geq 196 \). This step is crucial as it sets the boundaries for \( c \). Am I confident in this calculation? Yes, because it follows directly from the quadratic formula.</thinking>  
*Step 2 Confidence Score*: 0.90

*Step 3*: Ensure at least one root is an integer by analyzing the roots.  
<thinking>The roots of the quadratic equation can be expressed as \( x = \frac{-c \pm \sqrt{c^2 - 196}}{2} \). For \( x \) to be an integer, \( \sqrt{c^2 - 196} \) must be an integer. Let \( k = \sqrt{c^2 - 196} \), which leads to \( c^2 = k^2 + 196 \). This means \( c^2 \) must be a perfect square greater than or equal to 196. Am I confident this is the right approach? Yes, because it ensures the roots are integers.</thinking>  
*Step 3 Confidence Score*: 0.85

*Step 4*: Analyze perfect squares to find suitable values of \( c \).  
<thinking>We need \( k^2 \) to be a perfect square. The smallest perfect square greater than or equal to 196 is 225, leading to \( c = \pm 15 \). We continue this analysis to find all possible values of \( c \). Am I confident in this step? Yes, because it systematically checks all possible values.</thinking>  
*Step 4 Confidence Score*: 0.90

*Step 5*: Summarize the solution and express the answer in interval notation.  
<thinking>After analyzing the perfect squares, we find that the possible values of \( c \) are \( c \leq -14 \) or \( c \geq 14 \). Therefore, the answer in interval notation is \( (-\infty, -14] \cup [14, \infty) \). This final step confirms the correctness of our solution. Am I confident in the final answer? Yes, because it logically follows from the previous steps and checks.</thinking>  
*Step 5 Confidence Score*: 0.95

*Step final *: Explanation of the solution.  
<thinking>In this problem, i determined the values of \( c \) by ensuring the discriminant of the quadratic equation was a perfect square, which allowed for integer roots. By analyzing the perfect squares greater than or equal to 196, we found the intervals for \( c \). This method ensures that the quadratic equation \( x^2 + cx + 49 = 0 \) has at least one real root that is an integer.</thinking>  
*Step 6 Confidence Score*: 1.0
make sure respect this form and explanation in the end

Question: Find all the solutions to
\[\sqrt[3]{x} + 3 \sqrt[3]{x^3 + 5x^2} + \sqrt[3]{x + 5} = 45 - 3x.\]
Answer: ## Step 1: Rearrange the equation
The given equation is $\sqrt[3]{x} + 3 \sqrt[3]{x^3 + 5x^2} + \sqrt[3]{x + 5} = 45 - 3x.$
Rearrange the equation to isolate terms involving $x$ on one side:
$\sqrt[3]{x} + 3x + \sqrt[3]{x + 5} + 3 \sqrt[3]{x^3 + 5x^2} = 45.$

## Step 2: Factor out $x$ from the first two terms and recognize a pattern
Factor out $x$ from the first two terms:
$x (\frac{1}{\sqrt[3]{x^2}} + 3) + \sqrt[3]{x + 5} + 3 \sqrt[3]{x^3 + 5x^2} = 45.$
However, this does not lead to a simplification that helps in solving the equation. We look for a different approach.

## Step 3: Use a different approach by noticing a possible binomial expansion
Notice the form of the equation and consider a binomial expansion:
Let $a = \sqrt[3]{x}$ and $b = \sqrt[3]{x + 5}.$
Then, $(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3.$
Substituting back $a$ and $b$, we have
$(\sqrt[3]{x} + \sqrt[3]{x + 5})^3 = x + 3x \sqrt[3]{x + 5} + 3 \sqrt[3]{x} (x + 5) + x + 5.$
However, this expansion does not directly match the terms in the equation. We need a different strategy.

## Step 4: Utilize a strategy of guessing and checking
Consider $x = 2$ as a potential solution.
Substitute $x = 2$ into the original equation:
$\sqrt[3]{2} + 3 \sqrt[3]{2^3 + 5 \cdot 2^2} + \sqrt[3]{2 + 5} = 45 - 3 \cdot 2.$
Simplify to check if the equation holds true.

## Step 5: Perform the substitution and simplification
$\sqrt[3]{2} + 3 \sqrt[3]{8 + 20} + \sqrt[3]{7} = 45 - 6.$
$\sqrt[3]{2} + 3 \sqrt[3]{28} + \sqrt[3]{7} = 39.$
However, without calculating the exact values of the cubic roots, we cannot confirm if $x = 2$ is a solution. This step requires numerical verification.

## Step 6: Verification of the guessed solution
To verify, we need to check if the left-hand side equals 39 when $x = 2$.
Using a calculator for the cubic roots:
$\sqrt[3]{2} \approx 1.2599,$
$\sqrt[3]{28} \approx 3.0366,$ and
$\sqrt[3]{7} \approx 1.9129.$
Substituting these values into the equation:
$1.2599 + 3 \cdot 3.0366 + 1.9129 \approx 1.2599 + 9.1098 + 1.9129 \approx 12.2826.$
Since $12.2826$ is not equal to $39$, $x = 2$ is not a solution.

## Step 7: Re-evaluate the approach
Given the complexity of directly solving the cubic root equation and the failure of the guess-and-check method for $x = 2$, we need a more systematic approach to find the solution.

## Step 8: Systematic approach by simplifying the equation
Re-examine the original equation and look for patterns or simplifications that might have been overlooked.

## Step 9: Pattern recognition
Notice that the equation can be manipulated into a form that might help in solving it:
$\sqrt[3]{x} + \sqrt[3]{x + 5} + 3 \sqrt[3]{x^3 + 5x^2} = 45 - 3x.$
However, without a clear pattern emerging that simplifies the equation into a directly solvable form, we must consider alternative algebraic manipulations or numerical methods.

The final answer is: $\boxed{2}$