
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
Investigate the formula of \\(\cos nα\\) and draw the following conclusions:
\\(2\cos 2α=(2\cos α)^{2}-2\\),
\\(2\cos 3α=(2\cos α)^{3}-3(2\cos α)\\),
\\(2\cos 4α=(2\cos α)^{4}-4(2\cos α)^{2}+2\\),
\\(2\cos 5α=(2\cos α)^{5}-5(2\cos α)^{3}+5(2\cos α)\\),
\\(2\cos 6α=(2\cos α)^{6}-6(2\cos α)^{4}+9(2\cos α)^{2}-2\\),
\\(2\cos 7α=(2\cos α)^{7}-7(2\cos α)^{5}+14(2\cos α)^{3}-7(2\cos α)\\),
And so on. The next equation in the sequence would be:
\\(2\cos 8α=(2\cos α)^{m}+n(2\cos α)^{p}+q(2\cos α)^{4}-16(2\cos α)^{2}+r\\)
Determine the value of \\(m+n+p+q+r\\).

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