
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
Tim is participating in the following three math contests.  On each contest his score is the number of correct answers.
     $\bullet$  The Local Area Inspirational Math Exam consists of 15 problems.
     $\bullet$  The Further Away Regional Math League has 10 problems.
     $\bullet$  The Distance-Optimized Math Open has 50 problems.
For every positive integer  $n$ , Tim knows the answer to the  $n$ th problems on each contest (which are pairwise distinct), if they exist; however, these answers have been randomly permuted so that he does not know which answer corresponds to which contest.  Unaware of the shuffling, he competes with his modified answers.  Compute the expected value of the sum of his scores on all three contests.

*Proposed by Evan Chen*

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