
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
Let $z = (3m-2) + (m-1)i$, where $m \in \mathbb{R}$.
1. For what value of $m$ is $z$ a purely imaginary number? For what value of $m$ does the point corresponding to $z$ in the complex plane lie in the fourth quadrant?
2. If the coefficient of the third term in the expansion of $(1+2x)^m$ ($m \in \mathbb{N}^*$) is 40, find the value of $m$ and the corresponding value of the complex number $z$.

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