
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 the following two propositions:
p: For all $x \in \mathbb{R}^+$, the inequality $x \geq a\sqrt{x} - 1$ always holds;
q: The function $y = \log_a(x^2 - ax + 1)$ (where $a > 0, a \neq 1$) has a minimum value.
If among these two propositions, only one is true, then the range of the real number $a$ is __________.

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