
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
It is given that \( m, n>0 \) and \( m+n=1 \). If the minimum value of \( \left(1+\frac{1}{m}\right)\left(1+\frac{1}{n}\right) \) is \( a \), find the value of \( a \).

If the roots of the equation \( x^{2}-(10+a) x+25=0 \) are the square of the roots of the equation \( x^{2}+b x=5 \), find the positive value of \( b \).

If \( (x y-2)^{b-1}+(x-2 y)^{b-1}=0 \) and \( c=x^{2}+y^{2}-1 \), find the value of \( c \).

If \( f(x) \) is a polynomial of degree two, \( f(f(x))=x^{4}-2 x^{2} \) and \( d=f(c) \), find the value of \( d \).

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