
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 $a \in \mathbb{R}$, let the set $S = \{x \mid 2ax^2 - x \leq 0\}$ and $T = \{x \mid 4ax^2 - 4a(1-2a)x + 1 \geq 0\}$. If $S \cup T = \mathbb{R}$ (where $\mathbb{R}$ is the set of all real numbers), then the range of values for the real number $a$ is ______.

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