
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
Let  $S = \{(x, y) : x, y \in \{1, 2, 3, \dots, 2012\}\}$ . For all points  $(a, b)$ , let  $N(a, b) = \{(a - 1, b), (a + 1, b), (a, b - 1), (a, b + 1)\}$ . Kathy constructs a set  $T$  by adding  $n$  distinct points from  $S$  to  $T$  at random. If the expected value of  $\displaystyle \sum_{(a, b) \in T} | N(a, b) \cap T |$  is 4, then compute  $n$ .

*Proposed by Lewis Chen*

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