
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 projection of $\begin{pmatrix} 1 \\ 2 \end{pmatrix}$ onto a vector $\mathbf{u}$ is $\begin{pmatrix} 3/5 \\ 6/5 \end{pmatrix}$, find the projection of $\begin{pmatrix} 3 \\ -4 \end{pmatrix}$ onto $\mathbf{u}$. Additionally, find the projection of the result onto $\begin{pmatrix} 3 \\ 1 \end{pmatrix}$.

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