
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 the function $f(x) = \begin{cases} (4- \frac {a}{2})x & \text{for } 4x \leq 6 \\ a^{x-5} & \text{for } x > 6 \end{cases}$, where $a > 0$ and $a \neq 1$. If the sequence $\{a_n\}$ satisfies $n = f(n)$ and $a_{n+1} > a_n$ for $n \in \mathbb{N}$, determine the range of the real number $a$.

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