
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 $f(x)=(x+1)^{2}-3a\ln x$, where $a\in\mathbb{R}$.

$(1)$ Find the coordinates of the fixed point through which the graph of the function $f(x)$ passes;

$(2)$ When $a=1$, find the equation of the tangent line to the curve $f(x)$ at the point $(1,f(1))$ and the monotonic intervals of the function $f(x)$;

$(3)$ If for any $x\in[1,e]$, $f(x)\leqslant 4$ always holds, find the range of the real number $a$.

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