
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
11. Let  $f(z)=\frac{a z+b}{c z+d}$  for  $a, b, c, d \in \mathbb{C}$ . Suppose that  $f(1)=i, f(2)=i^{2}$ , and  $f(3)=i^{3}$ . If the real part of  $f(4)$  can be written as  $\frac{m}{n}$  for relatively prime positive integers  $m, n$ , find  $m^{2}+n^{2}$ .

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