
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
(Ⅰ) Given $z \in \mathbb{C}$, and $|z| - i = \bar{z} + 2 + 3i$ (where $i$ is the imaginary unit), find the imaginary part of the complex number $\frac{z}{2+i}$.

(Ⅱ) Given $z_1 = a + 2i$, $z_2 = 3 - 4i$ (where $i$ is the imaginary unit), and $\frac{z_1}{z_2}$ is a pure imaginary number, find the value of the real number $a$.

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