
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 3003rd, 3004th, and 3005th rows, respectively, with the leftmost element occurring at $i = 0.$  Compute
\[
2 \sum_{i = 0}^{3004} \frac{b_i}{c_i} - \sum_{i = 0}^{3003} \frac{a_i}{b_i}.
\]

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