
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
Observe the following series of expressions: $a+b=1$, $a^2+b^2=3$, $a^3+b^3=4$, $a^4+b^4=7$, $a^5+b^5=11$, ..., and so forth. When $n=11$, what is the value of $a^n+b^n=$ _____?

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