
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
A circle with diameter  $AB$  is drawn, and the point  $ P$  is chosen on segment  $AB$  so that  $\frac{AP}{AB} =\frac{1}{42}$  . Two new circles  $a$  and  $b$  are drawn with diameters  $AP$  and  $PB$  respectively. The perpendicular line to  $AB$  passing through  $ P$  intersects the circle twice at points  $S$  and  $T$  . Two more circles  $s$  and  $t$  are drawn with diameters  $SP$  and  $ST$  respectively. For any circle  $\omega$  let  $A(\omega)$  denote the area of the circle. What is  $\frac{A(s)+A(t)}{A(a)+A(b)}$ ?

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