
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
An eighth degree polynomial funtion  $ y \equal{} ax^8 \plus{} bx^7 \plus{} cx^6 \plus{} dx^5 \plus{} ex^4 \plus{} fx^3 \plus{} gx^2\plus{}hx\plus{}i\ (a\neq 0)$  touches the line  $ y \equal{} px \plus{} q$  at  $ x \equal{} \alpha ,\ \beta ,\ \gamma ,\ \delta \  (\alpha < \beta < \gamma <\delta).$  
Find the area of the region bounded by these graphs in terms of  $ a,\ \alpha ,\ \beta ,\gamma ,\ \delta .$ 

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