
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
The park is a square of $10 \times 10$ cells. You can place a lamp in any cell (but no more than one lamp in each cell).

a) The park is called well-lit if, no matter which cell a visitor is in, there is a $3 \times 3$ square that includes the visitor and contains at least one lamp. What is the minimum number of lamps for a well-lit park?

b) The park is called reliably well-lit if it remains well-lit even after any one lamp breaks. What is the minimum number of lamps for a reliably well-lit park?

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