
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 two propositions P and Q. P: For any real number $x$, $ax^2 + ax + 1 > 0$ always holds; Q: The equation $x^2 - x + a = 0$ has real roots. If $P \land Q$ is a false proposition and $P \lor Q$ is a true proposition, find the range of the real number $a$.

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