<Question>
Simplify $\frac{1+\sqrt{2}}{2+\sqrt{3}}$. Your solution can be converted to the form $A(1+\sqrt{B})-(\sqrt{C}+\sqrt{D})$, where $A$, $B$, $C$, and $D$ are positive integers. What is $A+B+C+D$?
</Question>

<think>
<Plan_1>
Okay, so I need to simplify the expression (1 + √2)/(2 + √3). Hmm, I remember that when you have a radical in the denominator, you can rationalize the denominator by multiplying the numerator and the denominator by the conjugate of the denominator. The conjugate of a binomial like a + b is a - b, right? So in this case, the denominator is 2 + √3, so its conjugate should be 2 - √3. Let me try that.

First step: Multiply both the numerator and the denominator by the conjugate of the denominator, which is 2 - √3.

<prompt_1.1> Multiply both the numerator and the denominator by the conjugate of the denominator, which is 2 - √3. </prompt_1.1>
</Plan_1>

<Execution_1>
<execution_1.1>
So, the expression becomes [(1 + √2)(2 - √3)] / [(2 + √3)(2 - √3)]. 

Let me compute the denominator first because that might be simpler. The denominator is a difference of squares: (2)^2 - (√3)^2 = 4 - 3 = 1. Wait, that's nice! So the denominator simplifies to 1.
</execution_1.1>
</Execution_1>

<Plan_2>
Based on execution_1.1:
If the denominator is 1, then the entire expression simplifies to just the numerator after multiplication. So, the numerator is (1 + √2)(2 - √3). Let me expand that.

<prompt_2.1> Expand the numerator (1 + √2)(2 - √3) and simplify. </prompt_2.1>
-----

Let me check if there's a way to express this in a different form that might have smaller integers, but given the problem statement, I think this is the required form. The problem says "Your solution can be converted to the form A(1 + √B) - (√C + √D)", so we need to present it in that specific form.
Our current expression is (2 - √3) + (2√2 - √2√3). Let's see if we can rearrange terms to match the given form.

First, note that √2√3 is √6, so we can write the expression as:

<prompt_2.2> Rearrange the terms to match the form A(1 + √B) - (√C + √D). </prompt_2.2>
</Plan_2>

<Execution_2>
<execution_2.1>
Using the distributive property:

1*2 + 1*(-√3) + √2*2 + √2*(-√3)

Calculating each term:

1*2 = 2

1*(-√3) = -√3

√2*2 = 2√2

√2*(-√3) = -√6

So combining all the terms in the numerator: 2 - √3 + 2√2 - √6
</execution_2.1>

<execution_2.2>
2 - √3 + 2√2 - √6.

Now, grouping the constant terms, the √3 terms, and the √2 and √6 terms:

= 2 + (2√2) + (-√3 - √6) - √6.

Wait, actually, let's list all the terms:

- Constant term: 2

- Terms with √3: -√3

- Terms with √2: 2√2

- Terms with √6: -√6

So the expression is 2 - √3 + 2√2 - √6.

Now, let's see if we can factor this into the form A(1 + √2) - (√3 + √6). Let's try to express 2 - √3 + 2√2 - √6 as A(1 + √2) - (√3 + √6).

Let's compute A(1 + √2):

A*1 + A*√2 = A + A√2.

Then subtract (√3 + √6):

A + A√2 - √3 - √6.

Comparing this to our expression 2 - √3 + 2√2 - √6, we can set up equations:

A = 2 (from the constant term)

A√2 = 2√2 (from the √2 term)

-√3 = -√3 (matches)

-√6 = -√6 (matches)

So, A must be 2. Therefore, the expression can be written as 2(1 + √2) - (√3 + √6).

Thus, in the form A(1 + √B) - (√C + √D), we have:

A = 2, B = 2, C = 3, D = 6.
</execution_2.2>
</Execution_2>

<Plan_3>
Based on execution_2.1:
Wait, the original expression was (1 + √2)/(2 + √3), and after multiplying numerator and denominator by (2 - √3), the denominator becomes 1, so the numerator is just (1 + √2)(2 - √3). So the entire expression simplifies to that numerator over 1, which is just (1 + √2)(2 - √3). But when I expanded the numerator, I got 2 - √3 + 2√2 - √6. So is that the simplified form? Let me check again.

<prompt_3.1> Verify the simplification process and confirm the result. </prompt_3.1>
-----

Based on execution_2.2:
Therefore, A + B + C + D = 2 + 2 + 3 + 6 = 13.

Let me check this again.

<prompt_3.2> Verify the final form and calculate A + B + C + D. </prompt_3.2>
</Plan_3>

<Execution_3>
<execution_3.1>
Wait, but maybe I made a mistake in expanding the numerator? Let me check each step again.

Original numerator: (1 + √2)(2 - √3). Let's expand this using FOIL:

First: 1*2 = 2

Outer: 1*(-√3) = -√3

Inner: √2*2 = 2√2

Last: √2*(-√3) = -√6

Adding them up: 2 - √3 + 2√2 - √6. That seems correct.

Denominator: (2 + √3)(2 - √3) = 2² - (√3)² = 4 - 3 = 1. That's correct. So the entire expression simplifies to 2 - √3 + 2√2 - √6.
</execution_3.1>

<execution_3.2>
Original expression: (1 + √2)/(2 + √3). Multiply numerator and denominator by (2 - √3):

Numerator: (1 + √2)(2 - √3) = 1*2 + 1*(-√3) + √2*2 + √2*(-√3) = 2 - √3 + 2√2 - √6.

Denominator: (2 + √3)(2 - √3) = 4 - 3 = 1.

So the simplified form is 2 - √3 + 2√2 - √6. Then, grouping terms:

2 + 2√2 - √3 - √6. Which can be written as 2(1 + √2) - (√3 + √6). Therefore, A=2, B=2, C=3, D=6. Sum is 2+2+3+6=13.
</execution_3.2>
</Execution_3>

<Plan_4>
Based on execution_3.2:
Wait, but let me confirm if this form is indeed what the problem is asking for. The problem says: "Your solution can be converted to the form A(1 + √B) - (√C + √D)", where A, B, C, D are positive integers. So yes, our expression 2(1 + √2) - (√3 + √6) is exactly in that form. Therefore, the values are A=2, B=2, C=3, D=6. Adding them together: 2 + 2 + 3 + 6 = 13.
</Plan_4>

<Execution_4>
No prompt to execute.
</Execution_4>

<Final_answer>
Therefore, $A+B+C+D = 2+2+3+6 = \boxed{13}$.
</Final_answer>
</think>