
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
A group of n  friends wrote a math contest consisting of eight short-answer problem  $S_1, S_2, S_3, S_4, S_5, S_6, S_7, S_8$ , and four full-solution problems  $F_1, F_2, F_3, F_4$ . Each person in the group correctly solved exactly 11 of the  12 problems. We create an 8 x 4 table. Inside the square located in the  $i$ th row and  $j$ th column, we write down the number of people who correctly solved both problem  $S_i$  and  $F_j$ . If the 32 entries in the table sum to 256, what is the value of n?

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