
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
Investigate the number of solutions for equation (1) with the variable $t$. Note that $t=1$ is a solution for any value of the parameter. According to the previous statements, this root does not correspond to any $x$. By Vieta's theorem, $t_{2}=\frac{a+2}{a+1}$. The original equation has two distinct roots if and only if:
\[
\left\{
    \begin{array}{l}
        \frac{a+2}{a+1}>2 \\
        \frac{a+2}{a+1}<-2
    \end{array}
\right.;
\quad
\left\{
    \begin{array}{l}
        \frac{a+2-2 a-2}{a+1}>0 \\
        \frac{a+2+2 a+2}{a+1}<0
    \end{array}
\right.;
\quad
\left\{
    \begin{array}{l}
        \frac{a}{a+1}<0 \\
        \frac{3 a+4}{a+1}<0
    \end{array}
\right.;
\quad
\left\{
    \begin{array}{l}
        a \in(-1 ; 0) \\
        a \in\left(-\frac{4}{3} ;-1\right)
    \end{array}
\right.
.
\]

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