
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 the function $f(x) = |4x(1-x)|$ defined on $(0, +\infty)$, if the equation $f^2(x) + (t - 3)f(x) + t - 2 = 0$ with respect to $x$ has exactly $3$ distinct real roots, then the set of values for the real number $t$ is \_\_\_\_\_\_.

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