
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
1. For the function $f(x) = x^3 + ax^2 + 3x - 9$, it is known that $f(x)$ has an extreme value at $x = -3$. Find the value of $a$.
2. Observe the following equations: $1 = 1$, $1 - 4 = -(1 + 2)$, $1 - 4 + 9 = (1 + 2 + 3)$, $1 - 4 + 9 - 16 = -(1 + 2 + 3 + 4)$, ... Based on this pattern, deduce the $n$th equation.

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