
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
A point $P(x_{0}, y_{0})$ is a fixed point on the curve $y = 3\ln x + x + k$ $(k \in \mathbb{R})$. The equation of the tangent line passing through point $P$ is $4x - y - 1 = 0$. Determine the value of the real number $k$.

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