
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  $C$  be the part of the graph  $y=\frac{1}{x}\ (x>0)$ . Take a point  $P\left(t,\ \frac{1}{t}\right)\ (t>0)$  on  $C$ .

(i) Find the equation of the tangent  $l$  at the point  $A(1,\ 1)$  on the curve  $C$ .

(ii) Let  $m$  be the line passing through the point  $P$  and parallel to  $l$ .  Denote  $Q$  be the intersection point of the line  $m$  and the curve  $C$  other than  $P$ . Find the coordinate of  $Q$ . 

(iii) Express the area  $S$  of the part bounded by two line segments  $OP,\ OQ$  and the curve  $C$  for the origin  $O$  in terms of  $t$ .

(iv) Express the volume  $V$  of the solid generated by a rotation of the part enclosed by two lines passing through the point  $P$  and pararell to the  $y$ -axis and passing through the point  $Q$  and pararell to  $y$ -axis, the curve  $C$  and the  $x$ -axis in terms of  $t$ .

(v)  $\lim_{t\rightarrow 1-0} \frac{S}{V}.$ 

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