
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
Utilize the properties of functions (such as monotonicity and oddity/evenness) to solve inequalities. Through the following set of problems, understand the applicable range of this method and what issues to pay attention to.

(1) Given the function $f(x) = x|x-2|$, find the solution set for the inequality $f(\sqrt{2} - x) \leq f(1)$.

(2) Given an odd function $f(x)$ defined on $\mathbb{R}$ that satisfies $f(x) = x^4$ for $x > 0$, and $f(x + t) \leq 4f(x)$ holds true for $x \in [1, 16]$, find the maximum value of the real number $t$.

(3) Given the function $f(x) = \begin{cases} 2, & x > 1 \\ (x-1)^2 + 2, & x \leq 1 \end{cases}$, find the solution set for the inequality $f(1 - x^2) > f(2x)$.

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