
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
Show the correctness of the following identities:

1. 

$$
\left|\begin{array}{ccc}
a & b & c \\
b+c & c+a & a+b \\
b c & c a & a b
\end{array}\right| \equiv-(a+b+c)(b-c)(c-a)(a-b)
$$

2. 

$$
\left|\begin{array}{lll}
b c & a & a^{2} \\
a c & b & b^{2} \\
a b & c & c^{2}
\end{array}\right| \equiv(a-b)(b-c)(c-a)(a b+a c+b c)
$$

3. 

$$
\left|\begin{array}{cccc}
1+a & 1 & 1 & 1 \\
1 & 1+b & 1 & 1 \\
1 & 1 & 1+c & 1 \\
1 & 1 & 1 & 1+d
\end{array}\right| \equiv abcd\left(1+\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}\right)
$$

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