
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Let \( x \) be the number of prize winners in only two subjects: mathematics and informatics, \( y \) be those in mathematics and physics only, \( z \) be those in physics and informatics only, \( a \) be the prize winners in all three subjects, and \( b \) be those only in physics. The system of equations is:

\[
\begin{aligned}
& \left\{
  \begin{array}{c}
    x + y + z + a + 20 = 40, \\
    y + a = 7, \\
    x + a = 10, \\
    z + a = 11, \\
    x + y + z + a + b + 20 = 51,
  \end{array}
  \Rightarrow
  \left\{
    \begin{array}{c}
      x + y + z + a = 20, \\
      y = 7 - a, \\
      x = 10 - a, \\
      z = 11 - a, \\
      b = 11,
    \end{array}
  \Rightarrow
  \right.\right. \\
& \left\{
  \begin{array}{c}
    10 - a + 7 - a + 11 - a + a = 20, \\
    y = 7 - a, \\
    x = 10 - a, \\
    z = 11 - a, \\
    b = 11,
  \end{array}
  \Rightarrow
  \left\{
    \begin{array}{c}
      a = 4, \\
      y = 3, \\
      x = 6, \\
      z = 7, \\
      b = 11.
    \end{array}
  \right.
  \right.
\end{aligned}
\]

Thus, the prize winners in physics are \( y + z + a + b = 25 \) individuals.

Solve the equation:

\[
x^{2} + y^{2} + 1 + \sqrt{4x^{2} + 4y^{2} - 34} = 2|x + y| - 2xy
\]

\[
\Rightarrow x^{2} + 2xy + y^{2} - 2|x + y| + 1 + \sqrt{4x^{2} + 4y^{2} - 34} = 0 \Rightarrow (x + y)^{2} - 2|x + y| + 1 + \sqrt{4x^{2} + 4y^{2} - 34} = 0 \Rightarrow (|x + y| - 1)^{2} + \sqrt{4x^{2} + 4y^{2} - 34} = 0
\]

(Since both terms are non-negative) \( (|x + y| - 1)^{2} = 0 \) and \( \sqrt{4x^{2} + 4y^{2} - 34} = 0 \Rightarrow \)

\[ 
\left\{
  \begin{array}{c}
    y = 1 - x, \\
    4x^{2} - 4x - 15 = 0, \\
    y = -1 - x, \\
    4x^{2} + 4x - 15 = 0,
  \end{array}
  \Rightarrow
  \left\{
    \begin{array}{l}
      x = 2.5, y = -1.5, \\
      x = -1.5, y = 2.5, \\
      x = -2.5, y = 1.5, \\
      x = 1.5, y = -2.5.
    \end{array}
  \right.
\right.
\]

Present the answer in LaTex format: \boxed{Your answer}
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