
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
In a monotonic increasing sequence of positive integers $\{a_n\}$, it is given that $a_1 = 1$, $a_2 = 2$, and $$(1+ \frac {a_{k}}{a_{k+3}})(1+ \frac {a_{k+1}}{a_{k+2}}) = 2$$ for any $k \in \mathbb{N^*}$. Find the value of $a_9$.

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