
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
If $\alpha, \beta$ are roots of $x^{2}-10 x+20=0$, find $a$, where $a=\frac{1}{\alpha}+\frac{1}{\beta}$.

If $\sin \theta=a\left(0^{\circ}<\theta<90^{\circ}\right)$, and $10 \cos 2 \theta=b$, find $b$.

The point $A(b, c)$ lies on the line $2 y=x+15$. Find $c$.

If $x^{2}-c x+40 \equiv(x+k)^{2}+d$, find $d$.

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