
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  $\triangle ABC$  be an equilateral triangle and let  $P$  be a point on  $\left[AB\right]$ .
 $Q$  is the point on  $BC$  such that  $PQ$  is perpendicular to  $AB$ .  $R$  is the point on  $AC$  such that  $QR$  is perpendicular to  $BC$ . And  $S$  is the point on  $AB$  such that  $RS$  is perpendicular to  $AC$ .
 $Q'$  is the point on  $BC$  such that  $PQ'$  is perpendicular to  $BC$ .  $R'$  is the point on  $AC$  such that  $Q'R'$  is perpendicular to  $AC$ . And  $S'$  is the point on  $AB$  such that  $R'S'$  is perpendicular to  $AB$ .
Determine  $\frac{|PB|}{|AB|}$  if  $S=S'$ .

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