
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
Define a sequence of complex numbers by $z_1 = 1 + i$ and
\[ z_{n+1} = z_n^2 + 1 \]
for all $n \geq 1$. In the complex plane, how far from the origin is $z_{5}$?

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