
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
For  $a>0$ , let  $l$  be the line created by rotating the tangent line to parabola  $y=x^{2}$ , which is tangent at point  $A(a,a^{2})$ , around  $A$  by  $-\frac{\pi}{6}$ .
Let  $B$  be the other intersection of  $l$  and  $y=x^{2}$ . Also, let  $C$  be  $(a,0)$  and let  $O$  be the origin.

(1) Find the equation of  $l$ .

(2) Let  $S(a)$  be the area of the region bounded by  $OC$ ,  $CA$  and  $y=x^{2}$ . Let  $T(a)$  be the area of the region bounded by  $AB$  and  $y=x^{2}$ . Find  $\lim_{a \to \infty}\frac{T(a)}{S(a)}$ .

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