
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 the complex number \( z \) satisfying
$$
\left|\frac{z^{2}+1}{z+\mathrm{i}}\right|+\left|\frac{z^{2}+4 \mathrm{i}-3}{z-\mathrm{i}+2}\right|=4,
$$
find the minimum value of \( |z - 1| \).

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