
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
(a) Find the maximum number of regions that can be formed by placing \( n \) lines in \(\mathbb{R}^{2}\).

(b) Find the maximum number of regions that can be formed by placing \( n \) planes in \(\mathbb{R}^{3}\).

(c) Find the maximum number of regions that can be formed by placing \( n \) circles in \(\mathbb{R}^{2}\).

(d) Find the maximum number of regions that can be formed by placing \( n \) spheres in \(\mathbb{R}^{3}\).

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