
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 function $y=\log_{a}x$ (where $0<a \ne 1$) has the inverse function $y=f^{-1}(x)$. There are four propositions about $f(x)$ and $f^{-1}(x)$. The correct propositions are:

1. Both functions necessarily have the same monotonicity.
2. When $a>1$, the graphs of the two functions have no intersection points.
3. If the graphs of the two functions intersect, the intersection must lie on the line $y=x$.
4. A sufficient but not necessary condition for the two functions to have intersection points is $0<a<1$.

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