
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
Among the following propositions:  
① The function $f(x) = \left(\frac{1}{3}\right)^x$ has a decreasing interval of $(-\infty, +\infty)$;  
② If the function $f(x) = \sqrt{x-1}$, then the domain of the function is $(1, +\infty)$;  
③ Given that the image of $(x, y)$ under the mapping $f$ is $(x+y, x-y)$, then the image of $(3, 1)$ under the mapping $f$ is $(4, 2)$.  
The correct proposition numbers are $\boxed{\text{①③}}$.

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