
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 \( f(x)=\left\{\begin{array}{lr}(3-a) x-3 & (x \leqslant 7), \\ a^{x-6} & (x>7),\end{array}\right.\) and the sequence \(\left\{a_{n}\right\}\) such that \(a_{n} = f(n)\) (\(n \in \mathbf{N}^{*}\)), and \(\left\{a_{n}\right\}\) is an increasing sequence, determine the range of the real number \(a\).

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