
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 \( a \) is the sum of all the positive factors of 2002, find the value of \( a \).

It is given that \( x>0, y>0 \) and \( \sqrt{x}(\sqrt{x}+\sqrt{y})=3 \sqrt{y}(\sqrt{x}+5 \sqrt{y}) \). If \( b=\frac{2 x+\sqrt{x y}+3 y}{x+\sqrt{x y}-y} \), find the value of \( b \).

Given that the equation \(||x-2|-1|=c\) has only 3 integral solutions, find the value of \( c \).

If \( d \) is the positive real root of the equation \(\frac{1}{2}\left\{\frac{1}{2}\left[\frac{1}{2}\left(\frac{1}{2} x^{2}+2\right)+2\right]+2\right\}=2\), find the value of \( d \).

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