
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 function $f(x)=3^x+k$ (where $k$ is a constant) and the point $A(-2k, 2)$ lies on the graph of the function $y=f^{-1}(x)$,

(1) Find the value of the real number $k$ and the expression for the function $f^{-1}(x)$;
(2) Translate the graph of $y=f^{-1}(x)$ by the vector $\mathbf{a}=(3,0)$ to obtain the graph of the function $y=g(x)$. If the inequality $2f^{-1}(x+\sqrt{m})-g(x) \geq 1$ always holds, determine the range of values for the real number $m$.

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