
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 functions $f(x)=3x-(a+1)\ln x$ and $g(x)=x^{2}-ax+4$. Find the range of real number $a$ for which the function $y=f(x)+g(x)$ is monotonically increasing within its domain. Determine whether there exists a real number $a$ such that the graph of the function $y=f(x)-g(x)$ is tangent to the x-axis. If such a number exists, find the number of values of $a$ that satisfy the condition and provide a justification.

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