
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
The 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 $4x-y-1=0$. What is the value of the real number $k$?

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