
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
Investigate the intersection points of the functions \( y = a^x \) and \( y = \log_a x \) when \( 0 < a < 1 \). Use the property of the function \( f(x) = x \ln x \): it is monotonically decreasing in the interval \( \left(0, \frac{1}{e}\right] \) and monotonically increasing in the interval \( \left[\frac{1}{e}, 1\right) \).

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