
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  $k$  be a positive real.  $A$  and  $B$  play the following game: at the start, there are  $80$  zeroes arrange around a circle. Each turn,  $A$  increases some of these  $80$  numbers, such that the total sum added is  $1$ . Next,  $B$  selects ten consecutive numbers with the largest sum, and reduces them all to  $0$ .  $A$  then wins the game if he/she can ensure that at least one of the number is  $\geq k$  at some finite point of time. 

Determine all  $k$  such that  $A$  can always win the game.

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