
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 $y=x+\frac{t}{x}$ with the following property: if the constant $t>0$, then the function is decreasing on the interval $(0,\sqrt{t}\,]$ and increasing on the interval $[\sqrt{t},+\infty)$.
(1) Given $f(x)=\frac{4x^2-12x-3}{2x+1}$, $g(x)=-x-2a$, $x\in[0,1]$, use the above property to find the monotonic intervals and the range of the function $f(x)$.
(2) For the functions $f(x)$ and $g(x)$ in (1), if for any $x\_1\in[0,1]$, there always exists $x\_2\in[0,1]$ such that $g(x\_2)=f(x\_1)$ holds, find the value of the real number $a$.

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