
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 sequence $\{b_n\}$ is an arithmetic sequence with the first term as -34 and a common difference of 1, and the sequence $\{a_n\}$ satisfies $a_{n+1} - a_n = 2^n$ ($n \in \mathbb{N}^*$), and $a_1 = b_{37}$, then the maximum value of the sequence $\left\{ \frac{b_n}{a_n} \right\}$ is \_\_\_\_\_\_.

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