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: Given that the graph of $|x| + |2y| = 10$ encloses an area of 100 square units, find the equation of a new graph that encloses an area of 50 square units and has a similar shape to the original graph, but is rotated 45 degrees counterclockwise.
Answer: The graph of $|x| + |2y| = 10$ is a rhombus with vertices at $(10,0)$, $(0,5)$, $(-10,0)$, and $(0,-5)$. Its area is 100 square units, which can be found using the formula $A = \frac{1}{2} \cdot d_1 \cdot d_2$ where $d_1$ and $d_2$ are the lengths of the diagonals.

To find the equation of a new graph that encloses an area of 50 square units and has a similar shape to the original graph, but is rotated 45 degrees counterclockwise, we need to find the vertices of the new rhombus.

Let the lengths of the diagonals of the new rhombus be $d_1'$ and $d_2'$. Since the area is halved, we have
\[ \frac{1}{2} \cdot d_1' \cdot d_2' = 50 \Rightarrow d_1' \cdot d_2' = 100 \]

Since the shape is similar, the ratio of the diagonals is the same:
\[ \frac{d_1'}{d_2'} = \frac{d_1}{d_2} = \frac{20}{10} = 2 \]

Solving for $d_1'$ and $d_2'$, we get:
\[ d_1' = 10\sqrt{2} \quad \text{and} \quad d_2' = 5\sqrt{2} \]

The vertices of the new rhombus are $(5\sqrt{2}, 5\sqrt{2})$, $(-5\sqrt{2}, 5\sqrt{2})$, $(-5\sqrt{2}, -5\sqrt{2})$, and $(5\sqrt{2}, -5\sqrt{2})$.

The equation of the new graph is:
\[ \boxed{|x + y| + |x - y| = 10} \]