
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
Let  $l$  be a tangent line at the point  $T\left(t,\ \frac{t^2}{2}\right)\ (t>0)$  on the curve  $C: y=\frac{x^2}{2}.$  A circle  $S$  touches the curve  $C$  and the  $x$  axis.
Denote by  $(x(t),\ y(t))$  the center of the circle  $S$ . Find  $\lim_{r\rightarrow +0}\int_r^1 x(t)y(t)\ dt.$ 

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