
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
Explore the pattern and solve the problem.  
$(1)$ Compare the sizes of $a^{2}+b^{2}$ and $2ab$ (fill in with "$ \gt $", "$ \lt $", or "$=$"):  
① When $a=3$, $b=3$, $a^{2}+b^{2}$______$2ab$;  
② When $a=2$, $b=\frac{1}{2}$, $a^{2}+b^{2}$______$2ab$;  
③ When $a=-2$, $b=3$, $a^{2}+b^{2}$______$2ab$.  
$(2)$ Based on the above fill-ins, speculate the relationship between $a^{2}+b^{2}$ and $2ab$, and explain the reason.

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