
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
Given any \( x \in A \) and \( y \in B \) (where \( A \subseteq \mathbf{R} \) and \( B \subseteq \mathbf{R} \)) such that there is a uniquely determined \( f(x, y) \), \( f(x, y) \) is called a bivariate function of \( x \) and \( y \).

Now, define a bivariate function \( f(x, y) \) on real numbers \( x \) and \( y \) as a generalized "distance" if it satisfies the following properties:
1. Non-negativity: \( f(x, y) \geq 0 \), and equality holds if and only if \( x = y \);
2. Symmetry: \( f(x, y) = f(y, x) \);
3. Triangle Inequality: \( f(x, y) \leq f(x, z) + f(z, y) \) for any real number \( z \).

Given three bivariate functions, select all that qualify as generalized "distances" for \( x \) and \( y \):
1. \( f(x, y) = |x - y| \);
2. \( f(x, y) = (x - y)^2 \);
3. \( f(x, y) = \sqrt{x - y} \).

The correct selection is: \(\qquad\)

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