
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 $\left\{a_{n}\right\}$ that satisfies:
$$
a_{1}=1, \quad a_{n}=5 a_{n-1}+3^{n-1} \quad (n \geqslant 2).
$$

Find $a_{n}$ and determine the maximum power of 2 that divides $a_{k}$ for $k=2^{2019}$.

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