
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 that the domain of $f(x)= \begin{cases} (2a-1)x+4, & \text{if } x\leq 1 \\ a^{x}, & \text{if } x > 1 \end{cases}$ is $\mathbb{R}$, and the sequence $\{a_n\} \, (n \in \mathbb{N}^*)$ satisfies $a_n=f(n)$ and $\{a_n\}$ is an increasing sequence, find the range of values for $a$.

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