
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 $a < 0$, the function $f(x)=a\cos x+ \sqrt {1+\sin x}+ \sqrt {1-\sin x}$, where $x\in\left[- \frac {\pi}{2}, \frac {\pi}{2}\right]$.
(1) Let $t= \sqrt {1+\sin x}+ \sqrt {1-\sin x}$, find the range of $t$ and express $f(x)$ as a function of $t$, denoted as $g(t)$;
(2) Find the maximum value of the function $f(x)$ (which can be expressed in terms of $a$);
(3) If for any $x_{1}$, $x_{2}$ in the interval $\left[- \frac {\pi}{2}, \frac {\pi}{2}\right]$, it always holds that $|f(x_{1})-f(x_{2})|\leqslant 1$, find the range of the real number $a$.

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