
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 sequence $\{a_n\}$ that satisfies $$a_{n}= \begin{cases} (3-a)n-3 ,  n\leq 7,    \
a^{n-6} ,  n>7\end{cases}(n∈N^{*})$$, determine the range of the real number $a$ if $\{a_n\}$ is an increasing sequence.

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