
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 \( Q \) be a plane passing through the origin. The vector \( \begin{pmatrix} 4 \\ 7 \\ 3 \end{pmatrix} \) when projected onto plane \( Q \), results in \( \begin{pmatrix} 1 \\ 3 \\ 1 \end{pmatrix} \). Determine the projection of the vector \( \begin{pmatrix} 3 \\ 1 \\ 4 \end{pmatrix} \) onto the plane \( Q \).

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