
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 $T_n$ be the $n^{th}$ Triangular number, defined by $T_1 = 1$ and for $n>1,$ $T_n = T_{n-1} + n$. Find the value of the product:
\[\prod_{k=2}^{10} \left( \frac{T_k}{T_{k-1}} - \frac{T_k}{T_{k+1}} \right).\]
Express your answer as a ratio $\frac{T_a}{T_b}$ for some integers $a$ and $b$.

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