
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
On a $2018 \times 2018$ grid, where a fly and $k$ spiders are moving. At each step, the fly moves to an adjacent cell (i.e., a cell that shares a side with its current cell) or decides to stay put; similarly, each spider also moves to an adjacent cell or decides to stay put.

(a) For which values of $k$ are the spiders guaranteed to catch the fly, regardless of the starting points of the $k+1$ creatures?

(b) Consider the same question if we replace the $2018 \times 2018$ grid with a $2018 \times 2018 \times 2018$ cube, where cells are replaced by cubes, with two cubes being adjacent if they share a face.

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