
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 function $f(x)= \begin{cases} (3-a)x+2 & \text{for } x \leq 2 \\ a^{2x^2-9x+11} & \text{for } x > 2 \end{cases}$, where $a>0$ and $a \neq 1$, let the sequence $\{a_n\}$ satisfy $a_n=f(n)$ for $n \in \mathbb{N}^+$, and $\{a_n\}$ is an increasing sequence. Determine the range of the real number $a$.

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