
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 $ f(x) $ is an even function defined on $ \mathbb{R} $, and for any $ x \in \mathbb{R} $, it holds that $ f(x-1) = f(x+3) $. When $ x \in [4,6] $, we have $ f(x) = 2^x + 1 $. If $ f^{-1}(x) $ is the inverse function of $ f(x) $ on the interval $ [-2,0] $, determine the value of $ f^{-1}(19) $.

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