
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
For real numbers  $a,\ b,\ c$ , on the coordinate plane with the origin  $O$ , let  $T$  be a linear transformation expressed by a matrix  $\left(
\begin{array}{cc}
a & 1 
b & c
\end{array}
\right)$ .

Suppose that the transformation  $T$  satisfies the following two conditions:

(i)  $(1,\ 2)$  is mapped to  $(1,\ 2)$ (ii) Denote  $A,\ B$  the images of  $(1,\ 0),\ (0,\ 1)$  by  $T$  respectively, the area of  $\triangle{OAB}$  is  $\frac 12$ .

Find the values of  $a,\ b,\ c$ .

*2011 Kyoto University entrance exam/Science, Problem 2*

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