
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
If the coordinates of vector $\overrightarrow{p}$ in a unit orthogonal basis $\overrightarrow{a}$, $\overrightarrow{b}$, $\overrightarrow{c}$ in space are $\left(1,3,2\right)$, then determine the coordinates of $\overrightarrow{p}$ in the basis $\overrightarrow{a}+\overrightarrow{b}$, $\overrightarrow{a}-\overrightarrow{b}$, $\overrightarrow{c}$.

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