
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
Consider the following equations:  
$\cos \frac{\pi}{4} + i\sin \frac{\pi}{4} = a_1 + b_1i$,  
$(\cos \frac{\pi}{4} + i\sin \frac{\pi}{4})^2 = a_2 + b_2i$,  
$(\cos \frac{\pi}{4} + i\sin \frac{\pi}{4})^3 = a_3 + b_3i$,  
…  
$(\cos \frac{\pi}{4} + i\sin \frac{\pi}{4})^n = a_n + b_ni$,  
where $i$ is the imaginary unit, and $a_n$, $b_n$ ($n \in \mathbb{N}^*$) are real numbers. By induction, the value of $a_{2015} + b_{2015}$ is \_\_\_\_\_\_.

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