
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
The graph of \( y = g(x) \) is described by the polynomial \( g(x) = \frac{(x-6)(x-4)(x-2)(x+2)(x+4)(x+6)}{720} - 5 \). The function \( g(x) \) is visualized on a coordinate grid with 1 unit between grid lines. Assume \( g(x) \) is defined only within the limits of the visible grid from \( x = -7 \) to \( x = 7 \).
Determine the sum of all integers \( c \) for which the equation \( g(x) = c \) has exactly 6 solutions.

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