
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
You are standing in an infinitely long hallway with sides given by the lines \( x = 0 \) and \( x = 6 \). You start at \((3,0)\) and want to get to \((3,6)\). Furthermore, at each instant, you want your distance to \((3,6)\) to either decrease or stay the same. What is the area of the set of points that you could pass through on your journey from \((3,0)\) to \((3,6)\)?

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