
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 ABC be a triangle with  $AB = 8$ ,  $BC = 9$ , and  $CA = 10$ . The line tangent to the circumcircle of  $ABC$  at  $A$  intersects the line  $BC$  at  $T$ , and the circle centered at  $T$  passing through  $A$  intersects the line  $AC$  for a second time at  $S$ . If the angle bisector of  $\angle SBA$  intersects  $SA$  at  $P$ , compute the length of segment  $SP$ .

*2016 CCA Math Bonanza Team #9*

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