
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
In a bag, there are 8 balls, labeled from 1 to 8. After stirring these 8 balls sufficiently, two balls are drawn from the bag at a time. Let denote  $ S$  the sum of the two balls.

(1) Find the probability such that  $ S$  doesn't exceed in eight.

(2) Let  $ f(n)$  be the probability such that  $ S$  takes value  $ n$ . Find the value of  $ n$  for which  $ f(n)$  is maximal.

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