
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
Given that \\( \begin{vmatrix} a & b \\\\ c & d\\end{vmatrix} =ad-bc\\), find the value of \\( \begin{vmatrix} 4 & 6 \\\\ 8 & 10\\end{vmatrix} + \begin{vmatrix} 12 & 14 \\\\ 16 & 18\\end{vmatrix} +…+ \begin{vmatrix} 2004 & 2006 \\\\ 2008 & 2010\\end{vmatrix} \\).

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