
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 $f(x) = kx^2 + (2k-1)x + k$, $g(x) = \log_2(x+k)$ ($k \in \mathbb{R}$)  
(1) If $f(0) = 7$, find the minimum value $m$ of the function $g(x)$ on the interval $[9, +\infty)$;  
(2) If $0 < g(1) \leq 5$, and the minimum value of the function $f(x)$ on the interval $[0, 2]$ is not less than $m$ from part (1), find the range of the real number $k$.

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