
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 vectors $\{\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}\}$ form a basis for a space, and the vectors $\{\overrightarrow{a}+\overrightarrow{b}$, $\overrightarrow{a}-\overrightarrow{b}$, $\overrightarrow{c}\}$ form another basis for the space, if a vector $\overrightarrow{p}$ has coordinates $(1,-2,3)$ in the basis $\{\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}\}$, find the coordinates of the vector $\overrightarrow{p}$ 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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