
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
(For this problem, full marks are 12 points.) Given $a>3$ and $a \neq 4$, consider the propositions P: "The exponential function $f(x) = (2a - 6)^x$ is monotonically decreasing on $\mathbb{R}$," and Q: "Both real roots of the equation $x^2 - 3ax + 2a^2 + 1 = 0$ are greater than 3." If either P or Q is true, but not both, find the range of the real number $a$.

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