
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 \( b(x) = x^{2} + x + 1 \). The polynomial \( x^{2015} + x^{2014} + \cdots + x + 1 \) has a unique "base \( b(x) \)" representation
\[ x^{2015} + x^{2014} + \cdots + x + 1 = \sum_{k=0}^{N} a_{k}(x) b(x)^{k} \text{,} \]
where:
- \( N \) is a nonnegative integer;
- Each \( a_{k}(x) \) (for \( 0 \leq k \leq N \)) is either the zero polynomial (i.e. \( a_{k}(x) = 0 \)) or a nonzero polynomial of degree less than \( \operatorname{deg} b = 2 \);
- The "leading digit \( a_{N}(x) \)" is nonzero (i.e. not the zero polynomial).

Find \( a_{N}(0) \) (the "leading digit evaluated at 0").

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