
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 proposition p: The function $f(x) = (2a-1)^x$ is an increasing function; proposition q: The function $y = \ln(2ax^2 - 2ax + 1)$ has the domain of $\mathbb{R}$, if $p \lor q$ is true, and $p \land q$ is false, find the range of the real number $a$.

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