
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 four propositions:  
\\(①\\) The negation of the proposition "If \\(a=0\\), then \\(ab=0\\)" is "If \\(a=0\\), then \\(ab \neq 0\\)";  
\\(②\\) The contrapositive of the proposition "If \\(q \leqslant 1\\), then \\(x^{2}+2x+q=0\\) has real roots";  
\\(③\\) If the proposition "\\(\neg p\\)" and the proposition "\\(p\\) or \\(q\\)" are both true, then the proposition \\(q\\) must be true;  
\\(④\\) The proposition "If \\(0 < a < 1\\), then \\(\log_{a}(a+1) < \log_{a}(1+ \frac{1}{a})\\)" is true.  
The correct proposition numbers are __________. (Fill in all the correct proposition numbers).

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