
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 soccer coach named  $C$  does a header drill with two players  $A$  and  $B$ , but they all forgot to put sunscreen on their foreheads.  They solve this issue by dunking the ball into a vat of sunscreen before starting the drill.  Coach  $C$  heads the ball to  $A$ , who heads the ball back to  $C$ , who then heads the ball to  $B$ , who heads the ball back to  $C$ ;
this pattern  $CACBCACB\ldots\,$  repeats ad infinitum.  Each time a person heads the ball,  $1/10$  of the sunscreen left on the ball ends up on the person's forehead.  In the limit, what fraction of the sunscreen originally on the ball will end up on the coach's forehead?

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