
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 a parabola with its vertex at the origin and focus at $F(1,0)$. Point $P$ is the symmetric point of $F$ about the $y$-axis, and the line passing through $P$ intersects the parabola at points $A$ and $B$.
1. Determine if there exists a point $T$ on the $x$-axis, distinct from $P$, such that the acute angles formed by $TA$, $TB$, and the $x$-axis are congruent. If such a point exists, find its coordinates. If not, provide a reason.
2. If the area of triangle $AOB$ is $\frac{5}{2}$, find the angle between vectors $\overrightarrow{OA}$ and $\overrightarrow{OB}$.

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