
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
Suppose that a function  $f(x)$  defined in  $-1<x<1$  satisfies the following properties (i) , (ii), (iii).

(i)  $f'(x)$  is continuous.

(ii) When  $-1<x<0,\ f'(x)<0,\ f'(0)=0$ , when  $0<x<1,\ f'(x)>0$ .

(iii)  $f(0)=-1$ Let  $F(x)=\int_0^x \sqrt{1+\{f'(t)\}^2}dt\ (-1<x<1)$ . If  $F(\sin \theta)=c\theta\ (c :\text{constant})$  holds for  $-\frac{\pi}{2}<\theta <\frac{\pi}{2}$ , then find  $f(x)$ .

*1975 Waseda University entrance exam/Science and Technology*

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