
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 an ellipse $E$ with its center at the origin $O$, two foci at $A(-1,0)$ and $B(1,0)$, and a vertex at $H(2,0)$.
1. Find the standard equation of the ellipse $E$.
2. For a point $P(t,0)$ on the $x$-axis, there exists a point $M$ on the ellipse $E$ such that $MP \perp MH$. Find the range of real numbers for $t$.

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