
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  $t$  be positive number. Draw two tangent lines to the palabola  $y=x^{2}$  from the point  $(t,-1).$  Denote the area of the region bounded by these tangent lines and the parabola by  $S(t).$  Find the minimum value of  $\frac{S(t)}{\sqrt{t}}.$ 

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