
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 $p$: The inequality $x + \frac{1}{x} \geqslant a^2 - a$ holds for any $x \in (0, +\infty)$; $q$: The equation $x + |x - 1| = 2a$ has real number solution(s). If $p \land q$ is true, find the range of values for the real number $a$.

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