
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  $ H$  be the piont of midpoint of the cord  $ PQ$  that is on the circle centered the origin  $ O$  with radius  $ 1.$ 
Suppose the length of the cord  $ PQ$  is  $ 2\sin \frac {t}{2}$  for the angle  $ t\ (0\leq t\leq \pi)$  that is formed by half-ray  $ OH$  and the positive direction of the  $ x$  axis. Answer the following questions.

(1) Express the coordiante of  $ H$  in terms of  $ t$ .

(2) When  $ t$  moves in the range of  $ 0\leq t\leq \pi$ , find the minimum value of  $ x$  coordinate of  $ H$ .

(3) When  $ t$  moves in the range of  $ 0\leq t\leq \frac {\pi}{2}$ , find the area  $ S$  of the region bounded by the curve drawn by the point  $ H$  and the  $ x$  axis and the  $ y$  axis.

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