
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
Independently flip a coin \( n \) times (where \( n \) is a positive integer), each flip resulting in either heads or tails. Let \( a(n) \) denote the number of ways for the coin to come up heads exactly a multiple of 3 times in \( n \) coin flips, and let \( b(n) \) denote the number of ways for the coin to come up heads exactly a multiple of 6 times in \( n \) coin flips.
1. Find \( a(2016) \) and \( b(2016) \).
2. For \( n \leqslant 2016 \), find the number of positive integers \( n \) such that \( 2b(n) - a(n) > 0 \).

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