
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
There are  $169$  lamps, each equipped with an on/off switch. You have a remote control that allows you to change exactly  $19$  switches at once. (Every time you use this remote control, you can choose which  $19$  switches are to be changed.)

(a) Given that at the beginning some lamps are on, can you turn all the lamps off, using the remote control?
(b) Given that at the beginning all lamps are on, how many times do you need to use the remote control to turn all lamps off?

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