
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
Compare $x^{2}+1$ and $2x$.  
$(1)$ Try (fill in with "$<$", "$=$", or "$>$"):  
① When $x=1$, $x^{2}+1 \_\_\_\_\_\_2x$;  
② When $x=0$, $x^{2}+1 \_\_\_\_\_\_2x$;  
③ When $x=-2$, $x^{2}+1 \_\_\_\_\_\_2x.$  
$(2)$ Induction: For any real number $x$, what is the relationship between $x^{2}+1$ and $2x$? Please explain the reason.

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