
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
Let $\overrightarrow {a_{1}} = 2\overrightarrow {m} - \overrightarrow {j} + \overrightarrow {k}$, $\overrightarrow {a_{2}} = \overrightarrow {m} + 3\overrightarrow {j} - 2\overrightarrow {k}$, $\overrightarrow {a_{3}} = -2\overrightarrow {m} + \overrightarrow {j} - 3\overrightarrow {k}$, $\overrightarrow {a_{4}} = 3\overrightarrow {m} + 2\overrightarrow {j} + 5\overrightarrow {k}$, find the values of real numbers $\lambda$, $\mu$, and $\nu$ such that $\overrightarrow {a_{4}} = \lambda \overrightarrow {a_{1}} + \mu \overrightarrow {a_{2}} + \nu \overrightarrow {a_{3}}$.

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