
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
Find the range of positive real numbers \( a \) such that the inequality \( \log_{a} x \leq x \leq a^{x} \) holds for all \( x \) in the interval \( (0, +\infty) \). Additionally, determine the values of \( a \) for which the graphs of the functions \( y = a^{x} \) and \( y = \log_{a} x \) intersect.

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