
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
Given a sequence $\{a_{n}\}$ that satisfies $2a_{n+2}+a_{n}=3a_{n+1}$, $a_{1}=1$, $a_{2}=5$, let $A(n,a_{n})$, $B(n,9)$, and $O$ be the coordinate origin. Find the maximum value of the area of $\triangle OAB$.

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