
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 diagram represents a grid of switches, each of which can be in one of two states: "on" and "off". Initially, all switches are off. Pressing any one switch will toggle its state and the state of all its adjacent switches. For example, pressing the switch at $(2,2)$ will toggle the states of the switches at $(1,2)$, $(2,1)$, $(2,2)$, $(2,3)$, and $(3,2)$. If each switch can be pressed at most once, we seek to minimize the total number of presses.

(1) If the final state requires exactly the switches at positions $(2,1)$, $(3,2)$, $(4,3)$, $(1,2)$, $(2,3)$, and $(3,4)$ to be on, what is the minimum number of presses required? Please provide a proof.

(2) If all switches are required to be off, what is the minimum number of presses required? Please provide the result and a solution.

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