
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
If a function $f(x)$ satisfies both (1) for any $x$ in the domain, $f(x) + f(-x) = 0$ always holds; and (2) for any $x_1, x_2$ in the domain where $x_1 \neq x_2$, the inequality $\frac{f(x_1) - f(x_2)}{x_1 - x_2} < 0$ always holds, then the function $f(x)$ is called an "ideal function." Among the following three functions: (1) $f(x) = \frac{1}{x}$; (2) $f(x) = x + 1$; (3) $f(x) = \begin{cases} -x^2 & \text{if}\ x \geq 0 \\ x^2 & \text{if}\ x < 0 \end{cases}$; identify which can be called an "ideal function" by their respective sequence numbers.

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