
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 the game of Colonel Blotto, you have 100 troops to distribute among 10 castles. Submit a 10-tuple  $(x_1, x_2, \dots x_{10})$  of nonnegative integers such that  $x_1 + x_2 + \dots + x_{10} = 100$ , where each  $x_i$  represent the number of troops you want to send to castle  $i$ . Your troop distribution will be matched up against each opponent's and you will win 10 points for each castle that you send more troops to (if you send the same number, you get 5 points, and if you send fewer, you get none). Your aim is to score the most points possible averaged over all opponents. 

For example, if team  $A$  submits  $(90,10,0,\dots,0)$ , team B submits  $(11,11,11,11,11,11,11,11,11,1)$ , and team C submits  $(10,10,10,\dots 10)$ , then team A will win 10 points against team B and 15 points against team C, while team B wins 90 points against team C. Team A averages 12.5 points, team B averages 90 points, and team C averages 47.5 points.

*2016 CCA Math Bonanza Lightning #5.4*

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