
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
Find the dimension \( d \) of the solution space, its basis (fundamental system of solutions), and the general solution of the homogeneous system of linear equations

$$
\left\{\begin{array}{r}
x_{2}+2 x_{3}-3 x_{4}=0 \\
2 x_{1}-x_{2}+3 x_{3}+4 x_{5}=0 \\
2 x_{1}+5 x_{3}-3 x_{4}+4 x_{5}=0
\end{array}\right.
$$

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