
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  $\{a_n\}_{n\geq 1}$  be a sequence defined by  $a_n=\int_0^1 x^2(1-x)^ndx$ .
Find the real value of  $c$  such that  $\sum_{n=1}^{\infty} (n+c)(a_n-a_{n+1})=2.$ 

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