
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 \( a, b \) be integers chosen independently and uniformly at random from the set \(\{0,1,2, \ldots, 80\}\). Compute the expected value of the remainder when the binomial coefficient \(\binom{a}{b}=\frac{a!}{b!(a-b)!}\) is divided by 3. (Here \(\binom{0}{0}=1\) and \(\binom{a}{b}=0\) whenever \(a<b\)).

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