
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 $i$ is the imaginary unit, and the real part of a complex number $z$ is denoted as $\text{Re}(z)$, for example, if $z=-2+3i$, then $\text{Re}(z)=-2$. Given that the complex number $z=1+i$, a student performed the following calculations:
$$z^2 = (1+i)^2 = 2i, \quad \text{Re}(z^2)=0$$
$$z^3 = (1+i)^3 = -2+2i, \quad \text{Re}(z^3)=-2$$
$$z^4 = (1+i)^4 = -4, \quad \text{Re}(z^4)=-4$$
$$z^5 = (1+i)^5 = -4-4i, \quad \text{Re}(z^5)=-4$$
Based on this, it is inferred by induction that $\text{Re}(z^{2017})$ is equal to:
A) $2^{2017}$
B) $-2^{2017}$
C) $2^{1008}$
D) $-2^{1008}$

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