
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
Given that the function $f(x) = \frac{(x+1)(x+a)}{x^2}$ is an even function,
(1) Find the value of the real number $a$;
(2) Let the set $E = \{y | y = f(x), x \in \{-1, 1, 2\}\}$, $\lambda = (\lg 2)^2 + \lg 2 \lg 5 + \lg 5 - \frac{1}{4}$, determine the relationship between $\lambda$ and $E$;
(3) When $x \in \left[\frac{1}{m}, \frac{1}{n}\right]$ ($m > 0, n > 0$), if the range of the function $f(x)$ is $[2 - 3m, 2 - 3n]$, find the values of $m, n$.

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