
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 $n \geq 3$ be a prime number, and let $E(n)$ be the number of triplets $(b_1, b_2, b_3)$ of distinct integers with $1 \leq b_i \leq n$ for all $i$ such that $b_1 + b_2 + b_3 = n+2$. Determine $E(n)$.

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