
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
In every node of an infinite grid paper, a vertical pin is placed. A needle of length \( l \) lies on the paper parallel to the grid lines. For which \( l \) can the needle be rotated by \( 90^{\circ} \) without leaving the plane of the paper? The needle is allowed to move freely on the plane as long as it passes between the pins; the thickness of the pins and the needle is negligible.

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