
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 positive integer is said to be 'good' if each digit is 1 or 2 and there are neither four consecutive 1's nor three consecutive 2's. Let \( a_{n} \) denote the number of \( n \)-digit positive integers that are 'good'. Find the value of \(\frac{a_{10} - a_{8} - a_{5}}{a_{7} + a_{6}}\).

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