
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 that $f(x)$ is a function defined on the set $M$, if $D \subseteq M$, and for any ${{x}_{0}} \in D$, it follows that $f({{x}_{0}}) \in D$, then the function $f(x)$ is said to be closed on the interval $D$.

(1) Determine whether $f(x)=x-1$ is closed on the interval $\left[-2, 1\right]$, and provide a rationale for your answer.
(2) If the function $g(x)=\frac{3x+a}{x+1}$ is closed on the interval $\left[3, 10\right]$, find the range of possible values for the real number $a$.
(3) If the function $h(x)={{x}^{2}}+2x$ is closed on the interval $\left[a, a+1\right]$, find the range of possible values for $a$.

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