
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
If $a,b \in (0,+\infty)$ and $a+b=2$, then the following inequalities are always true for all $a,b$ that satisfy the conditions. Fill in the blank with all the correct inequality numbers.

$(1) ab \geqslant 1$;  
$(2) \sqrt{a} + \sqrt{b} \leqslant \sqrt{2}$;  
$(3) a^2 + b^2 \geqslant 2$;  

$(4) \frac{1}{a} + \frac{1}{b} \geqslant 2$.

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