
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
Shown below are rows 1, 2, and 3 of Pascal's triangle.

\[
\begin{array}{ccccccc}
& & 1 & & 1 & & \\
& 1 & & 2 & & 1 & \\
1 & & 3 & & 3 & & 1
\end{array}
\]
Let \((a_i),\) \((b_i),\) \((c_i)\) be the sequence, from left to right, of elements in the 10th, 11th, and 12th rows, respectively, with the leftmost element occurring at \(i = 0.\) Compute
\[
\sum_{i = 0}^{10} \frac{b_i}{c_i} - \sum_{i = 0}^{9} \frac{a_i}{b_i}.
\]

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