
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 every  $0 < \alpha < 1$ , let  $R(\alpha)$  be the region in  $\mathbb{R}^2$  whose boundary is the convex pentagon of vertices  $(0,1-\alpha), (\alpha, 0), (1, 0), (1,1)$  and  $(0, 1)$ . Let  $R$  be the set of points that belong simultaneously to each of the regions  $R(\alpha)$  with  $0 < \alpha < 1$ , that is,  $R =\bigcap_{0<\alpha<1} R(\alpha)$ .

Determine the area of  $R$ .

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