
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 $m\ge 5$ be an odd integer, and let $T(m)$ denote the number of triples $(a_1, a_2, a_3)$ of distinct integers with $1\le a_i \le m$ for all \$i\$ such that the sum of these integers is exactly $m$. Find the value of $T(m)$ for a given odd integer $m$.
A) $\frac{m(m-1)}{6}$
B) $\frac{m(m-2)}{6}$
C) $\frac{m(m-3)}{6}$
D) $\frac{m(m-4)}{6}$

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