
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 $f(x)=ax+ \sqrt {x+1}$, where $a\in R$.
(1) When $a=1$, find the minimum value of $f(x)$;
(2) If the points on the graph of the function $f(x)$ are all in the plane region represented by the inequality group $\begin{cases} x+1\geqslant 0 \\ x-y-1\leqslant 0 \end{cases}$, find the range of the real number $a$;
(3) If the function $h(x)=x^{4}+[f(x)- \sqrt {x+1}](x^{2}+1)+bx^{2}+1$ has zero points in $(0,+∞)$, find the minimum value of $a^{2}+b^{2}$.

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