
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
Square $PQRS$ has a side length of 2 units. Points $T$ and $U$ are on $PQ$ and $SQ$, respectively, such that $PT = QU$. When the square is folded along the lines $RT$ and $RU$, sides $PR$ and $SR$ coincide and align with the diagonal $RQ$. Determine the value of $PT$ that can be expressed in the form $\sqrt{k}-m$ units and find the integer value of $k+m$.

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