
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
Let the function $f(x)=\begin{cases} (3-a)x-3 & (x\leqslant 7), \\ a^{x-6} & (x > 7), \end{cases}$ and the sequence $\{a_n\}$ satisfy $a_n=f(n)$, $n\in\mathbb{N}^*$, and the sequence $\{a_n\}$ is increasing. Determine the range of the real number $a$.

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