
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 the following five propositions:
① The graph of the function $y=f(x)$, where $x \in \mathbb{R}$, may intersect with the line $x=a$ at two different points;
② The function $y=\log_{2}x^{2}$ and the function $y=2\log_{2}x$ are equal functions;
③ For the exponential function $y=2^{x}$ and the power function $y=x^{2}$, there always exists $x$, such that for $x > x$, $2^{x} > x^{2}$ holds;
④ For the function $y=f(x)$, $x \in [a, b]$, if $f(a) \cdot f(b) < 0$, then $f(x)$ has a zero point within $(a, b)$.
⑤ Given $x_{1}$ is a root of the equation $x+\lg x=5$, and $x_{2}$ is a root of the equation $x+10^{x}=5$, then $x_{1}+x_{2}=5$.
Among these, the correct propositions are.

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