
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 the tangent line at the point  $P(s,\ t)$  on a circle  $C:x^2+y^2=1$ . Denote by  $m$  the line passing through the point  $(1,\ 0)$ , parallel to  $l$ . Let the line  $m$  intersects the circle  $C$  at  $P'$  other than the point  $(1,\ 0)$ .

Note : if  $m$  is the line  $x=1$ , then  $P'$  is considered as  $(1,\ 0)$ .

Call  $T$  the operation such that the point  $P'(s',\ t')$  is obtained from the point  $P(s,\ t)$  on  $C$ . 

(1) Express  $s',\ t'$  as the polynomials of  $s$  and  $t$  respectively.

(2) Let  $P_n$  be the point obtained by  $n$  operations of  $T$  for  $P$ .

For  $P\left(\frac{\sqrt{3}}{2},\ \frac{1}{2}\right)$ , plot the points  $P_1,\ P_2$  and  $P_3$ .

(3) For a positive integer  $n$ , find the number of  $P$  such that  $P_n=P$ .

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