
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
Let  $L$  be a positive constant. For a point  $P(t,\ 0)$  on the positive part of the  $x$  axis on the coordinate plane, denote  $Q(u(t),\ v(t))$  the point at which the point reach starting from  $P$  proceeds by　distance  $L$  in counter-clockwise on the perimeter of a circle passing the point  $P$  with center  $O$ .  

(1) Find  $u(t),\ v(t)$ .

(2) For real number  $a$  with  $0<a<1$ , find  $f(a)=\int_a^1 \sqrt{\{u'(t)\}^2+\{v'(t)\}^2}\ dt$ .

(3) Find  $\lim_{a\rightarrow +0} \frac{f(a)}{\ln a}$ .

*2011 Tokyo University entrance exam/Science, Problem 3*

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