
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 $f(x)=\begin{cases} & (3-a)x-3,x\leqslant 7, \\\\ & {{a}^{x-6}},x>7. \\end{cases}$, if the sequence ${a_{n}}$ satisfies $a_{n}=f(n)(n∈N^{*})$, and ${a_{n}}$ is an increasing sequence, determine the range of values for the real number $a$.

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