
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 that $z = \log_2(1 + m) + \log_{\frac{1}{2}}(3 - m)$ where $m \in \mathbb{R}$,

1. Find the range of $m$ if the point corresponding to $z$ in the complex plane lies in the third quadrant.
2. Find the value of $m$ if the point corresponding to $z$ in the complex plane lies on the line $x - y - 1 = 0$.

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