
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
If \( f(x) = a_0 + a_1 x + a_2 x^2 + \cdots + a_4 \cos x^{4028} \) is the expansion of \( \left(x^2 + x + 2\right)^{2014} \), then find the value of
\[ 
2a_0 - a_1 - a_2 + 2a_3 - a_4 - a_5 + \cdots + 2a_{4020} - a_{4027} - a_{4028}
\]
.

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