
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 vector $\mathbf{v}$ such that
\[\operatorname{proj}_{\begin{pmatrix} 3 \\ 1 \end{pmatrix}} \mathbf{v} = \begin{pmatrix} \frac{45}{10} \\ \frac{15}{10} \end{pmatrix}\]and
\[\operatorname{proj}_{\begin{pmatrix} 1 \\ 4 \end{pmatrix}} \mathbf{v} = \begin{pmatrix} \frac{65}{17} \\ \frac{260}{17} \end{pmatrix}.\]

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