
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  $n$  be a positive integer.  Let  $(a, b, c)$  be a random ordered triple of nonnegative integers such that  $a + b + c = n$ , chosen uniformly at random from among all such triples.  Let  $M_n$  be the expected value (average value) of the largest of  $a$ ,  $b$ , and  $c$ .  As  $n$  approaches infinity, what value does  $\frac{M_n}{n}$  approach?

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