
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
Define a parabola  $C$  by  $y=x^2+1$  on the coordinate plane. Let  $s,\ t$  be real numbers with  $t<0$ . Denote by  $l_1,\ l_2$  the tangent lines drawn from the point  $(s,\ t)$  to the parabola  $C$ .

(1) Find the equations of the tangents  $l_1,\ l_2$ .

(2) Let  $a$  be positive real number. Find the pairs of  $(s,\ t)$  such that the area of the region enclosed by  $C,\ l_1,\ l_2$  is  $a$ .

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